QA 306 

.P3 



■ 



. 



■ 



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*• 




Class JSA^l 
Book ^ 
GopyrigtitN . 

COPYRIGHT DEPOSIT. 



ERRATA. 

/ sign of division. 

Index 37'th line, read: General dis- 
cussion (Px) and the line, pp. 93- 
109. Integral law, etc. 

Spell integer, pp. 50, 51. 

P. (91), line (7), read: pp. 9, 10, 
65-, and p. (65), at bottom page. 

P. (92), line (4), read p. (88) . 

At the fcot of title page read: 
Copyright applied for, 1911, by O. A. 
Palmer, the Author. 



PALMER'S STUDIES 

IN 

LIMITS 




By Rev. O. A. Palmer, 

Copyright Applied For. 



CIA297492 



►v 



PREFACE 



The Author of these studies is a retired minister of 
the Gospel, of the Congregational Church, having, at the 
time these studies were begun, closed a work of continued 
pastorates extending over a period of forty years. Im- 
paired health unfitted him for thorough pastoral work 
and it was thought best to retire from this field of activity. 
The mind of the Author went back to his College life 
at the Ohio Wesleyan University at Delaware, Ohio. He 
was again in the class room with Professor Perkins, lost 
in the deep soundings of the calculus and hidden meaning 
of the differential, the infinitesimal. An ambition to 
find this North Pole of the higher Mathematics began 
a study which has lost the Author to home and the 
world for the past six years. The work itself tells 
how much, and, comparatively, how little, has been ac- 
complished. The proofs given of Fermat's Theorem are 
only corollaries, incidents, and not at all the purpose of 
these studies. The Author did not know of Fermat's 
Theorem until he had been pursuing these studies for 
four years. The purpose, at first, was to find the place 
and meaning of the infinitesimal or differential. After- 
ward he finds the general function containing the laws 
of all magnitudes and limits. These limits are all con- 
tained at zero, the universal limit. At this general limit, 
the Author finds the content of magnitude, and a way to 
locate the general limit. These studies are pure theory. 
The presumption is that the above general function and 
its treatment applies to the laws of all forces. 

These studies are only suggestive of what yet lies 
beyond their vision. They are at every step incomplete, 
and but meagerly illustrated. Age and other considera- 
tions have led the Author to lay down the task of further 
thought on a theme which seems to him to promise un- 
limited rewards to the student. 

Better to the Author of these studies, than the meager 
results of his work, has been the companionship of the 
Father who has thought and wrought with his child in 
all the very feeble efforts of his life. 

THE AUTHOR. 
Enid, Oklahoma. 
July 1st, 1910. 



A magnitude will approach zero by reducing the size 
of its unit of measure (1), as (1) yard, (1) square foot, 
etc. It may become absolute zero by the negative rela- 
tions of the terms that express portions of the magnitude. 

Infinitesimal zero approaches absolute zero as its 
limit. If we find an expression in which these two limits 
agree for all magnitudes, then will this expression contain 
all magnitudes and limits. For the relations of the lines 
of an infinitesimal magnitude are the same as those of 
the magnitude at any size. 

If a magnitude, as (Fx), have a measure (x), sl 
collection of units, so that (x) measures from some fixed 
point all its dimensions, and (y) represents a distance 
between two limits of (x) , so that (x-y)=0 numerically 
when (y) = (x) ; and, (x-y)=zero when the unit of (x) 
and (y) become infinitesimal, (numerically unchanged), 
then will F(x-y) become zero on either above hypothesis, 
and F(x-y) on the infinitesimal hypothesis approaches 
F(x-y) on the hypothesis of absolute zero. 

Therefore F(x-y) is absolute zero on the hypothesis 
the unit of (x) and (y) become infinitesimal. 

F(x-y), numeral and magnitude, holds the relation of 
(x) and (y) variable; x n =nx n - 1 , y n =nx n - 1 , as shown far- 

y x 

ther on ; and, on this hypothesis expressed in this differ- 
ential law, F(x-y), (magnitude and number) =zero. 

I give this introduction to my study so as to give the 
purpose that runs through the discussion of the theme. 

DEFINITION (1). 

A measure is a unit, or a collection of units considered 
concrete, applied to a magnitude to show the relation of 
the magnitude to the measure. 

Remarks. 

Unity in this study is not number and has no power 
as such. 

The size of any length, surface or volume, depends 
upon its variable unit of measure, and not upon the 
number of such units. 

Absolute unity is the last and absolute hypothesis. 

—5— 



If (a) represents a number of variable units of mag- 
nitude, and (b) represents a different number of the 
same unit, then if the unit become infinitesimal will 
a=b as symbols of magnitude. Let the magnitude be a 
line, as: ABC, AB=(a), BC=(6). Suppose 
the unit to diminish, and (A) and (C) to approach (B). 
Then, when the unit is zero, (a) = (6) as magnitudes. 
Both equal zero. As numbers, concrete, they have the 

a concrete a 
same ratio as before. — = — discrete. It is now 

b concrete b 
a ratio. 




Let (ABC) be any line, or symbol of any magnitude. 
Take (D), (E), and (F), points on the line (ABC), 
marking portions of (ABC) that lie between the points 
(D), (E), and (F). To fix the points (D), (E), and 
(F), assume arbitrarily the point (0). Any other point 
might be taken. Draw through (0) the rectangular lines 
(YY) (XX) for the purpose of relating all points in 
(ABC) to these lines, and so adjust our measure to the 
magnitudes. The law of the magnitude fixes the relation 
of its points to each other. The rectangular lines relate 
these points to (0). Draw (EE 1 ) and (FF 1 ) perpen- 
dicular to (XX). Then (OE 1 ) and (OF 1 ) will be the 
distance of the points (E) and (F) from (YY). Denote 
the distance between (EE 1 ) and (FF 1 ) by (y), and so, 
generally, let (y) = distance between points on (ABC). 
Also let (^ , ) = (0F 1 ) or distance of any point from (0) 
on (XX). Distances to the left of (O) will be negative; 
distances to the right of (0) will be positive. When 
(x) = (OF 1 ), then (OE 1 ) will be (x-y). (x-y) will fix 
the point (F 1 ) from (E 1 ). Now, if we assume the 
general line (EE 1 ), (FF 1 ), (DD 1 ) = (r), when (x) be- 
comes (x-y), the value of the ordinate (x n ) will be 
(x-y) n =x° — nx™- 1 y+n (n-l)x n - 2 y 2 y n . 

1.2 
• —6— 



Now If we transpose (x n ) and divide by (y) we have 

(x-y) n — # n = — nx*- x -\-n (n-1 ) x n ~ 2 y, etc. 

y ~¥ 

Now, (pp. 5 and 6), let the common unit of (x) 
and (y) become infinitesimal; then, in the second member 
of the above equation, the terms containing (y) will be 
zero, as (y) is taken to represent the units of magnitude 
and the other factors represent discrete numbers. The 
first member of the equation is a ratio, (pp. 5 and 6), 
Also the first term of the second member. Then 
( x-y ) n — x n = — nx*- 1 . 

y ~y 

Now put (y) = (x) as magnitudes, which they are 
by hypothesis, and (x-y) n =0, and we have -\-x n =-\-nx n - 1 t 

or, x n =nx n - 1 . If the limit be (x-\-y) it approaches (x) 

aT 
by subtracting (y) from (y). If the limit is (x-y) then 
it approaches (x) by adding (y). The variations are 
negative to each other so that the ratio is negative to (y). 
(+y) requires ( — nx*- 1 ), ( — y), (+nx n ~ x ). (x+y) or 
(x — y) are points from which (x) f the required limit, is 
reached. This is done in the above series by making 
(y)=0r (—y) + (+y), (+y) + (—y) as above. (x-y) 

x n 
becomes (x). (x+y) becomes (x). — =nx xl - 1 is not the 

x 
ratio of (x) to (x n ) f but the constant ratio of change of 
the ratio x n . When (y) is distance between two limits of 

aT 
(x), then x R =nx n - 1 . This was shown above. For, (x±y) n 

Y V 

becomes zero on the hypothesis that the unit of (x) and 
(y) was infinitesimal, on this hypothesis alone we had left 
— x n = — nx nl . That is, the rate or ratio of the distance 

y 

between any two limits and the limit (x n ) of the function 
is (±nx nl ) when the magnitude or function approaches 
(x) at zero. 

Referring to the figure (p. 6), (OF) n =N(OF 1 ) nl . 

The above is based on the theory that the line (ABC) 
in all its proportions is conceived to occupy an infinitesi- 
mal space. Each limit (y) is, in its unit, infinitesimal, 

—7— 



and the line (ABC) exists, in its beginning, inappreciably 
small, in all its relations as to its parts. The magnitude 
is then asumed to be at any limit just what it is at zero 
infinitesimal. So what is true at the infinitesimal is 
always true. 

A little more algebraically expressed, F(x±y) is the* 
limit from which we approach (Fa;). Now, as (x±y) 
approaches (x), F(x±y) approaches (Fa;). Using again 
The figure on (p. 6), as the unit of (x) and (y) dimin- 
ishes, all points on (ABC) approach (0), and (Fa:) and 
F(x±y) are one magnitude at zero. But, also, F(x±y) 
approaches (Fa:) by changing (±y) numerically. In the 
variable rate, x^^nx*- 1 we have seen that the rate of 

y 
change must be, for (+2/), ( — nx n - 1 ), and the reverse, when 
x n =0. This we will now show by the formula. 

- (1) Fx^(F i x)(F n _ 1 y)J r (F 2 x)(F n _ 2 y)-\-(F 3 x)(F n _ 3 y) .. 
-\-Fy. Differentiating the above, we have: 

,.(2)F 1 x+F 1 x+-(F 2 x) (F n _ l2 /) + (F 2 a) (F n _ !? y) + (F 3 x)(F^ 2 y) 
-f (F 3 a0 (F n _ 2 y) etc. If we assume ( — y) then F^ is plus, 
series (1). If we assume (~+y) then F^ is minus in 
series (1), so that the terms in (J) are alternately ( + ) 
and (^-), and series (2) consists of two identical parts, 
and one part is negative and the other positive; so that 

series (1) is constant at zero. 

The hypothesis that (±y) is the distance between the 
limits of (x), and (±x) a distance between limits of 
(y) , fixes a relation between (a*) and (y) that adds two 
Conditions to the series of two independent variables. 
N There are two series in the series (2). Each series is on 
the hypothesis that one variable is constant, (x) and (y) 
are units of measurement. In differentiating (,xyz) y 
[(x) and (y) constant], (z) is the unit of measure, and 
its differential is therefore (1) ; and we have xy-\-.xz-\-yz. 
The differential of . each variable is its unit, and in this 
the variables are alike. 

The variable (Ax) is superfluous and an unknown 
quantity in calculations of the calculus. It is a great hin- 
drance, mathematicians could not define. The usual way. 
to get rid of it is to catch an opportunity to divide the 
entire expression by it. This is not always at hand, 
especially when there are several variables (a-), (y) , and 
(2), when there would appear with each its (A a*), (Ay), 
(Az); etc. (Ax), (Ay) is (1), when each is the unit of 
measure. The differential law for (Fx) and (Fy) above 
is, based on the hypothesis that (x-y) approaches the 

—8— 



limit (x), and (y-x) approaches the limit (y) . This is 

% an hypothesis of limit that applies to all magnitudes, 

whether measured, as in the illustration (p. 6), by 

rectangular co-ordinates or by polar angle. x n ==nx n ~ 1 . 

■ y 

y n =ny n - 1 . We came to this form by making the unit of 

x 

the variables infinitesimal. Also at the same time (ABC) 
became infinitesimal zero. The numerical or ratio value 
of (x) and (y) are still independent. x n =±nx n ~ 1 , y n = 

y x 

ztnx 11 - 1 . For (x-y) and (y-x) are = numerically. Now, 

if we make (y) numerically equal to (x) , (this page), 
then x^==nx n - x , also x n =nx n - 1 . But (y) is the differential 

V x 

and so is (x) and they have the same result, (nx^ 1 ). 

They are equal, (dif. a;) = (dif. y) on a natural hypo- 
thesis; that is, that (x~y) = (x), at its limit, and, (y-x) 
~=(y), at the limit of (y-x). 

To illustrate, let y=5 and x=3; then (3-5) approaches 
(3) by the difference or differential, ( — 2). 

This is true if a third variable is introduced. It would 
sustain a like relation to (x) and (y) . We would have: 
[(x-j-y) ±z~\ n or (xy±z) n . Fx,y+(F x x,y) (F n _ x z)etc. Series 
(1) , (p. 8) . This would give two series of identicals with each 
letter or six different series included in the above devel- 
opment of (Fx, y) with (z) distance betweeii ■ limits, 
using also (x) and (y) as measures of limits. 

ILLUSTRATION. 

That the above disuse of (Ax), (Ay) is correct, and 
that (y) is a symbol as above: — take the equation x*-\-x 2 — : 
41a: — 105. The equation and its derived functions are: 

(x*+x 2 — 41a;— 105), (3x 2 +2x— 41), (6a;+2), (6). 

Now, if we develop this (Fx) as in formula (1, p. 8), 
using (y) as distance between limits of (x) , observing 
(F n _ 1 y) = (y), (F n . 2 y) = (if) , (F n _,y) =if, etc. 

(2 ) 1.2.3 

(x*+x 2 — 41a;— 105) + (3z 2 +2x-41)i/+ (6z-f2) y*+ (6)^ 

2 2.3 

Dividing by factorials (2) and (2.3), 

(x z +x 2 — 41a-— 105) + (Bx 2 +2x— 41)2/+ (3a;+l)?/ 2 -M/ ! . 

On (pp. 8 and 9) we have shown series (1) (p. 8)=0. ; 
The (Fa;) is therefore equal to the remainder of the 

series (1). But in the equation given above, (Fa;)=zpro, 

—9— 



therefore the remainder of the series (l)=zero. And 
' Sz : — 2; j -41 i y — (Sx — 1 1 ;/■ — y=Q, for that^ limit to which 
(z-y approaches. 

The roots of the equation are ( — 3), ( — 5), (-f-7). 
If the limit — be taken for (x), and ( — 3)=(x-y) f 
then i i i== — i — 2 i=2. Substitute this value for (y) above: 

z : — 8j=3o. -r : — Sz— 16=35 — 16=121 
3 3 3 9 3 9 9 

*— 4==^ 121 — — 11 *=U— 4=- J_or — 15_or — 5. 

3 9 3 3 3 3 3 

It will be seen that we reach only the limit — 5 and 
series '1 p. S > is equal to zero f:r every separate 
limit of < z i when the corresponding limit is (z-u). The 
limit ( — 5 > would be found by taking the distance between 
(— 7) and i — 5). 7= < z-y >=— 5 — (—12) =7. v=12. 

If 12 be substituted above i — -5) will result. 

If (z-u) reaches the limit (7) then — 5=(x-y). — 5= 
(+7— 12)=— 5. (y) = (— 12 i for the limit (7). (y) 
will be r— 2). (-2). (—12). (-12). (—10). (-10). 

In the above work differentia! i z i is unity, It will be 
seen that (y>. :^ : . j^ : . etc.. is a differentia! series reversed. 
2 2.3 

The function is \, ' . No symbol i_y' nor (_y) : , 

1.2.3. . .?i 
etc.. comes intc this series naturally. It has never been 
though: necessary. Yet it is a series of derived functions 
of (y), just the same as the series of (Fjt) with which 
it comomes :n series (1> ana :n Tayior s Tneorem. I: 
■_ ■• and (_;-i : etc. are necessary in the one part 
of Taylor's Theorem they are in the ether. That they 
are not different in value from i_y) is evident in Tay- 
lor's Theorem because no differentia! for (>) in y- 

1.2.3. . .■<■: 

is introduced. 

In the above problem (Ax) and (Ax) 2 , (A*)- would 
have been involved, and no way to determine their sig- 
nificance. If we put them in the place of ':/) as distance 
between limits of < ;.' ) . then they are powers of this 
difference. 

Tie sum of series (1) (p. 8), is zero with the hypo- 
thesis that (x) is distance between limits of (y), and (y) 
is the distance between limits of id. Taylor's Theorem 
is a specif.c form of series '!'. and its sum is also zer r 
when it denotes limits of (Fx'i. 

—10— 



ANALYSIS OF (Fx) OR (Fy). 

(x-y) approaches all values of (x) in (Fx). If so, then 
series (1), (p. 8) contains all the limits of (Fx)=x n . 

(x-y) is any limit of (Fx) ; and (x) a limit or any limit 
to which (x-y) is equal when (y) is differential. (x-y) n 
is (Fx) as illustrated in derivingj^ n =n£ n -\ (pp. 6, 7). 

V 

Now we wish to show that (x) n differentiated and 
combined with (y) integrated is the development of 
(x±y) n . If we differentiate (x) n successively we have 
x n -\-nx n - 1 -j-n(7i — l)x n ~ 2 -\-n(n — 1) (n — 2)x n -\ etc. 

Integrate (1) or (y) and we have y, y 7 , y» t y k , . . .yr_ 

2 23 1.2.3.4 L.tt 
or the differential is negative so there will be a change 
of signs above. This has been already shown (pp. 8, 9). 

General (Fx)=Ax n -\-Bx n - 1 -{-Cx n - 2 . . . .L. In developing 
(Ax n ) we can multiply the whole development series (1) 
(p. 8) by (A). Series (1) includes (Ax n ) . Likewise 
series (1), (p. 8), includes (Bx nl ) , for (n-1) may be 
made = (t) or (n). All the properties of series (1) 
(p. 8) will be true of (Bx nX ), and so with each succeeding 
term of (Fx). Each will be equal to zero and the sum 
of all will be zero. So (Fz)=Az n -f-Ba; n - 2 -f-Ca; n - 3 . . . +L 
is included in series (1). This will be shown to be so 
when (Fi/)=A 1 2/ n -t:B 1 2/ n -'+C , 2/ n - 2 . . . .+L 1 . 

Series (1), (p. 8), includes (y) n . The second term has 
[i/n — ( n1 )], or (y). But (A 1 ) will then be a factor 
of (Fa;) in series (1) and the series (y n ) developed will 
have the factor (A 1 ) and will be equal to zero, and con- 
tain all the limits of (Fx) with (y) as distance between 
the limits. Now if (Fx) developed as in series (1), (p. 8) 
be applied to (B l 2/ nl ), we will have the result zero, and 
so with all the terms of (Ft/)=A 1 ^"-f-B 1 y n ~* . . . . L 1 . 
Then the sum of these series will be equal to zero. The 
constants (L) and (L J ) will disappear in all the terms 
except the (Fx) and (Fy) and must be omitted from these 
terms to make series (l)=zero. When (L) and (L 1 ) are 
variables they are not omitted. The law of this complex 
series (1) is that (x-y) approaches (x). This limit being 
reached, the series is zero. The series must be zero to 
reach the limit (x) . When (Fy)=A* iT-fB 1 y n - x . . . L l , 
then the values of (y) are related to each other by the 
law of (Fi/)=A 1 y n -{-B 1 y n \ etc. (y-x) approaches (y) 
and series (1), (p. 8)=zero on this hypothesis. With the 
complex functions two laws, one fixing values of (x) , 

—11— 



and the other fixing values of (y) are made to harmonize 
with each other, so that (x) is a general limit for [y) 
and (y) for (x), (y-x) approaches x-y) approaches 

(x), when series (1) is zero. 

i F.v)=A.r r — Bx*- 1 .... L gives [x) an unlimited num- 
ber of values when L is variable. 

So also (Fy)=A 1 r^-B 1 y n ~ , . . .L. Series (1) 
contains all limits. If (Fa;) contain the form [x a : ) , 
then, this form being incommensurable must be considered 
apart. 

As we have already shown that each element or term 
of a function [x) cr (Fyj when united in series (1) 
(p. 8), with (Fy), or (F.r). respectively, gives the series 
Q)=zero. So that we have (F.r ab ).and series I F a b) — 
(F : ;rab) (F^ l2 f),ete. 

This series must approach zero for {x-y) approaches 
(x). 

If we have (F.ra t>). then (Fy) is also incommensurable, 
for. going back to figure (p. 6). put I 'OF i = : — 
and (OE) = (a— V&), then —y=\ — 2 \ b) . Both (x) 
and (y) have the same incommensurable unit — \ b. 
(x) and [y) are measured by the same unit, \ 6. the 
unit of \6. As (x-y) approaches (x), both (x) and 
(y) becomes zero (p. 6). If (Fac) be taken less than '1' 
then its limits will be less than (x). The derived func- 
tions of (Fa*) will proceed rapidly to zero, and the series 
may be said to end when a term is reached that is too 
small for practical consideration. 

INTEGRALS. 
Series (1). ip. S. 

F?+(F x x) ir,;')-iF.^ <F ._; -F t) (J^i I . . F; = 
zero. Fx= the remainder of the series. When (F 
is known. (Fx) will be found by obtaining the successive 
derived functions of (.r), and so completing the series. 
This is a universal integral law. When we have (F.x), 
in which the first derived function is incommensurable 
with (.''). then the series is infinite. But we may take 
(Fx) less than (x) and :he derived functions approach 
zero. The smaller I Fx can be taken, the more rapidly 
the series approaches the zero. For when a derived 
function is reached that is inappreciable, then the next 
term is infinitesimal cr zero. Let (;) be the [Ft 
Then F. :=the differential of the magnitude. 

In the arc of a circle I F. : > = I 1- - ) ' -. — : 1 1-;- 1 =F : : 

—12— 



[+(l-a 2 )- 3 ' 2 +3a; 2 (l-# 2 )- 5,2 ]=F 3 z. 

% '[9(l-a; 2 )- 5/2 +90a; 2 (l-z 2 )- ?/2 +105^(l-£ 2 ) -°' 2 ]=-F 5 z. 
[225z(l-a: 2 )- 7 / 2 +1050ar' (l-x 2 )- 0/2 +M5x r '(l-x 2 )-' l/ 2]=F G z. 
[225(l-a- 2 )- 7/2 +4725^(l"^ 2 )-° 2 +14175^ 4 (l-^-)" 11/2 +10395x a 
(1-x 2 ) ~^ 2 =F 7 z. 

[11025a; (1-aO - f,/2 +99225a; 2 (l~a:n -"/ 2 +218295s 5 (l-a; 2 )- 13 ' 2 + 
[135135a; 7 (l-x 2 )-* 5/2 '\=F s z. 
[11025 (1-a; 2 )- 9 ' 2 etc.]=F 9 z. 

(y), in series (1), (p. 8), is not the coordinate (y) in 
the rectangular coordinates of the circle. Some other 
letter must be used to represent that, since, in the series 
(1) and in the above problem (y) is distance between 
limits of (x) in the (Fz). This is remarked so that there 
may be no confusion caused by the usual use of (y) to 
represent the ordinate (EE), Figure (p. 6.) 

Now if we put (x')=zero in the problem (p. 12) and 
apply the terms of (Fy),(F^ x y)=y, (F n - 2 2/)==#j etc., and 

~2 
derive the law of the numerals, we have y + j/ 3 +3.3?/ 5 

1.2.3 1..5 
+3.3.5.5. y " +3.3.5.5.7.7. y» +3.3.5.5.7.7.9.9. t/ 11 + 

1...7 1...9 1...11 

3 2 .5 2 .7 2 .9 2 .11 2 y 1 * etc. 

1...13 

Now (y) is the differential of (x), and we may put 
it=to any value of (x) and we have x-\- a; 3 + 3 2 a; 5 + 3 2 .5 2 

1.2.3 1..5 

a: 7 3 2 .5 2 .7 2 x 9 + 3 2 .5 2 .7 2 .9 2 x 11 . etc. 



1..7 1..9 1..11 

If we take (a;) %» it=sin. of the arc of 30°. These 

terms then become .5 =(F X z) (F n 1 y) 

.02083333= (F 3 z) (F ns y) 
.00234375= (F 2) (F n a y) 
.00034877= (F 7 z) (F n 7 y) 
.00005933= (F 9 z) (F n9 y) 
.00001092= (F,,z) (F n _ lx y) 
.00000211= (F 13 z) (F n _ 13 y) 

.52359821= (Fz) .X 6=3.14159826=* 
The above series at the top of the page is when a , = 1 /^, 
1 (2) 3 (3) 5 (4) 7 (5) 9 (6) 11 



2.3.2 4.5.2 6.7.2 8.9.2 10.11.2 12.13.2 
etc. 

In this abbreviation the numbers in parenthesis stand 

—13— 



for the number of the term. The successive terms are 
obtained by multiplying as indicated. This will shorten 
the work of series at the head of the page. If (z) be the 
segment of a circle, then the differential of area of seg- 
ment is (1 — x 2 ) 12 . The series forms similar to that on 
(p. 13), for a portion of the arc of the circle. The first 
term is plus, the remaining terms minus. The last factor 
is not doubled and we have y — y 2 — Sy 5 — 3.3.5i/ 7 — 



3 2 .5 2 .7y°, —etc. 



1..3 1..5 1..7 




Make (2/)=# =1 /2 for 30° and we have the terms: 



.5 



— .02083333= 

— .00078125= 

— .00006974= 

— .00000847= 

— .00000121= 



F x z 



F„ * | 

Fn-3 V\ 
Fn- 5 2/| 
Fn- 7 y\ 
Fn- 9 #| 

F^IF^.i/l 

.47830600=z=CDEF, (above figure.) 
— .2165063 =CEF. 



F 3 z 
F 5 z 

F 7 z 

F z 



.2617997 =CDE. 
12 
3.1415964 =Area of the circle when (x) is % of R. 
The diameter is (2), the series (1), a?=% 
or measure in series (1). 

y(l+dy 2 ) 1/2 =the differential of a surface of revolution. 
(y) is here the ordinate (EE 1 ) (p. 6). It is not the 
differential {y) . 

For the sphere it will be 27r(R 2 -z 2 ) 1/2 (I^ )=2 1 rR(F l a?) 

(R 2 — x) 

=R. 2?r is a factor of the series or integral. 

—14— 



Then (F^) (F n _ 1 2/)=%. Make (y) = ( x ) and we have (Rx). 
x=R and the result is for the surface of the sphere. 
27r2/(l+<fy 2 ) 1 ' 2 ~27rR 2 =surface of */ 2 the sphere. 4ttR2= 
surface of sphere when {x) is =(R), as on (p. 14). tt= 
3.14159 the area of the circle when R/2=% and the diam- 
eter is (2). This shows that (x) has been a unit of 
measure of (Fx), (Ax) has not been an element. ir(y 2 ) = 
differential of a solid of revolution, (y) here is the or- 
dinate (p. 6). In the circle or sphere y 2 =R 2 — x 2 . 
ff (R*— **)=*> *(&— x2)y= (F lZ ) (F n _ l2 /). 
n(—2x)=F 2 z.—2*py 2 /2=F 2 zF n 2 y. 
tt(— 2)=F 3 z.— 2tt y±==F 3 z. F n _ 3 y. 

Make x=0. 

irWy—ir ^ =Fz=solid. Put (i/) = (x) = (R). (i/) in 
3 
(F n _!2/) is not the ordinate (2/). 
ttR 3 — ttR 3 = 2jrR 3 — i/ 2 sphere, 4/3 ^R3 _ Volume. 

3 3 

(d)=diameter=2R. d 3 =8R 8 . R s =d». 

8 
ird* ==4/3ttR 3 = Volume of sphere. 

(jry 2 ) for the paraboloid is ir(2px). 

^pxy^F^Fn^y. 

*2py 2 =F 2 xF n _ 2 y. 

2 
x=0 and (y) = (x) and we have 7rpx 2 =volume of para- 
boloid. 

In all these integral problems we have used (y) to rep- 
resent the ordinate, and afterward following the notation 
of (y) as a distance between limits of (x) in series (1). 
No confusion has come of it; as the ordinate (y) is always 
substituted in the series by (Fx).(y), the ordinate disap- 
pears at the beginning of the solution. l^=Fx. — 1 

1+x (1+x)' 

=F x x. 2_ = F 2 x. 2.3 =F 3 x. Fx+F x x F n _^+F 2 ic 

(1+x) 3 (1+x)* 

F n _ 2 ?/+F 3 x F n _ 3 2/= 1 — 1 y+ __J 2/ 2 — 1 2 / 3 

1+x (1+x) 2 (1+x) 3 (1+x) 4 
Put x=0, and 1 — 1/+2/ 2 — y s , etc.=series, and — ( 1 ) = 

(l+xT~ 
— x+x 2 — x 3 , etc., an infinite series. If we separate this 
into two series, so as to take the difference we have: 

—15— 



1 =x-\-x z -\-x 3 , etc., ad infinitum, — x- — a* 4 — x e , etc. 
1— x 

Multiply the first terms of these series by the ratio, 
proceeding from the right to the left .rxl x-=l x. Now 
the first terms to the right are infinite and equal and their 
subtraction will not change the result as we are taking 
the difference between the two parts of the series. (A — ...) 
— (B— ...)=A— B. 

i (i) i a—x) x 

Then : 1 = •-_ == = first series. 

x (x) x ( x ) 1 — x 3 

1 —1 —1 —x'- x x- 

X s \/(x 2 —l) 1—X 2 1— *2 l—x'2 l— X 2 

X 2 

X X' 2 X 



1 X s 1—X 

Such expressions as ( b ) having (x 1 ), have no vari- 

(a-x) 
aoie limit as (y). If (Fa*) is (x), then (F x x) is (1). 
F^F^ y=(y). The series ends with (y), and (y) has no 
variable power. But on the hvpothesis that (x-y) alwavs 

1 ' 
=x, (y) has no existence. The series F xF n 1 y=± — 

a±x) 

etc., (p. 15) has no existence, and the series (1), in this 
case, is Fx= b d(x) or (_x) has no variable 

(a—x) 
power. (dx) or (Lx) in Taylor's Theorem should be 
omitted as it duplicates the meaning of (y). All develop- 
ments of b in series, by division or otherwise, gives a 

a±x 

sum with (x) in both numerator and denominator which 
shows that _5_ is the quotient. The rules of development 

a±x 
do not apply to this, as the development is simply the first • 
term b . The series does not include the first power 

a — x 
of a variable of the form (x^-1). b , by division gives, 

a — x 
b bx bx 2 bjr bx 4 
h a — x= — — — ± — — — ± — = etc 
a a 2 a 1 a 4 a 1 

—16— 



b ba ba- ba 5 ba l 

x — a=± — -p — ± — -p — ± — =p etc. 



*As *\- *(.' **/ «X/ 



By the rule or law of division both of these series are 
the same algebraic quotient. The series are alike except 
(a) and (x) exchange places. When we make (a) the 
measure or divisor the first quotient is constant and the 
remainder of the series all variable. When the variable 
is the measuree or divisor the entire series of terms are 
variable. When the first series is infinite, the second series 
approaches zero, and the reverse. Either may be infinite 
according as (x) is greater or less than (1). 

If a, b, c, d, e, etc., be limits of (x), then (fr-a)=distance 
between these limits, (a) and (b). Then (x-y)=x — (b-a). 
If this expression is limit (a), then (x) — (b-a)=a. x= 
(b-a)-\-a=b. If x — (b-a) marks the limit (b) and ap- 
proaches (a), then x — (a-b)=b in the expression (x-y) 
in series (1), (p. 8). Then (x)=a-b-\-b=a. All tlies e 
limits of- (x) may be thus compared. If the constant (L) 
be added to the general (Fx)=Ax n +Bx n - 1 -\-Cx n - 2 , then 
series (1), (p. 8) will not be zero unless (L) be a variable 
and =Fx, or (Fx) = — (L). In series (1), (L) disap- 
pears except in (Fa*). This series of derived functions 
of (x) and (y) does not contain values of (#). It ap- 
proaches all limits of (x) . It is a differential and integral 
series. It holds all values of (Fx) while it remains at 
zero in its variables (x) and (y) and in its general form. 

The symbols A> B, C, L, in (Fx), are related to each 
other as derived functions of (L). These, as related to 
each other, are variables. Their units of measurement are 
the limits of (x), — (a, b, c, etc.). 

Let (Fx 1 )^=x n ^-A 1 x n +B <x n ~ 2 +C l x n ^ — L =(x-a) (x-b) 
(x-c) etc. — Fx=x nJ r (a-{-b-\-c-\-d etc.)^ 11 - 1 ^- (afr+ac+ad-f 
bc-\-bd-\- cd)x n - 2J r (abc-\-abd-\-bcd etc.) a; 11-3 -{-... (abode etc.). 

Let us take the function of the fourth degree. Then(F#) 
=x*-\- (a-\-b-\-c-{-d) x 3 . 

\ab\ \ahc\ 

\ac\ \acd\ 

\ad\x- '--\-\bcd\x-\-abcd. 

\bc\ \abd\ 

\bd\ 

\cd\ 

(Fa; 1 ), for this law of coefficients, is (L 1 )=abcd==(Fx'). 
With (a) variable we have F^x=(bcd), (b) variable gives 
(acd) and so on, (abd), abc, 

—17— 



acd\ 
abd\ 



\ac\bc\ 
\ad\cd\ 



\ac 
lab 



a\ 
b\ 
c|2.3.4. 

d\ 



24. 



bcd\gives\cd\bd\=2\bc 2.3 
abc\ a6|a6| \cd 

adlacl \ad\ 
bd\bc\ \bd\ 
The differentials of a, b, c, d, etc., are unity, always plus. 
( — a) has a negative power. As a measuring unit or 
differential it is (+1). (—a) (—6) (+c) (+d)=L J 
=FxK 

abc ab\ 

abd — ad\ 
-bed 
— acd 



2.3 



-b 
c 

d\ 



2.3.4.=+24. 



— ac 
—bo 
—bd 
cd 
In differentiating, a, b, c, etc., as constants, keep their 
signs as when alone. Each has its own sign and not the 
sign of the combination. 

Thus +abc gives (+a&), ( — ac), ( — be). In the above 
factors (x-a), (x-{-c) etc., (x) = (a). (x) = ( — c) when 
(FaO=0. 

In this analysis of (F#) there is but one limit consider- 
ed, that is (a), or (b), or (c). (Fx) is not necessarily= 

zero. It is not so if (x~a), (x-b) are considered only 

as factors of (Fa;). There will be as many binomial factors 
as there are units in the degree of the function. Only one 
limit is considered or expressed by (Fx). 




OD=(a) for the limit (D 1 ). OC=(b) for the limit 
(C 1 )- (%) does not approach from one limit to another 
by the difference between limits. (Fx) is not zero until 
we change the sign of (a) and consider (x-a) to approach 
(-f-a) from the origin, or (0), in the figure above. 

Then (#-a)=zero when the limit of (x) is (a), (x-a) 

—18— 



is not only a factor of (Fx) , but (x) is limited to (+a). 
The signs of the limits (a), (b) , (c) , etc., in the factors 
(x±a), (xztb), are negative to the values of the limit. 

If (Fx) x =^L\ then (Fx) will be r+A'r-'-fB'x 11 - 2 . . . 
+L'= (Fx) l + (F^) * (F n _^) + (F,z) ' (F n _ 2 x) + (F s x) ' 
(F n _ 3 o:) - - -+Fz. (F 1 ^) 1 =K; (F 2 o:) 1 =2J, etc. In this 
series (Fa:) at the close is the basal element of (Fo:)=a: n + 

A^^+B's 11 - 7 L< It is (x n ). Then we have for the 

series L+(F 1 L)aj4-r i L) x* +(F 3 L) a^+(F 4 L) _x^ 

2 2.3 2.3.4 

+ (F n L) ( a* ). 

(l./.n) 

This is o: n -f A^^-f-B'o: 11 - 
order. 



etc., with terms in reverse 



The following may present a concise view of the pre- 
ceding analysis of the series (1), (p. 8). The general 
function of the fourth degree will illustrate. 



4.3.2.1 


3 


2 


a 




ab\ \abc 




|1.2.3.4 


a: 4 + 




b 


x z +2 


ac\ \abd 








c 


2.3 


ad\x 2 -\-l\acd 


x-\-abcd 




\d\ \bc\2 \bcd 


1 




\bd\ 




\cd\ 


( 4a: 3 + 


Sx 2 + 2x + 1)2/. 


(12a: 2 -f 


6x + 2)y 2 . 




2 


(24a: 


+ 


6 


): 


A 







2.3 



(24) y< 



2.3.4. 



The perpendicular columns under a: 4 , x z , x 2 , x, including 
a: 4 , x 3 , x 2 , x, have each its respective multiplier in the dif- 
ferential horizontal series at the top. If (y) be taken 
minus, the sum of each perpendicular column is divisible 
by (x-y) and by the hypothesis that (x-y) approaches 
(x) each column=zero and so the sum of all, leaving 
(L) out,= zero. 

If we differentiate the series of coefficients A 1 , B 1 , C\ 
D\ L\ we have L+(F, L)x+(F 2 L)^ 2 _ +(F 3 L) x z + 

2 2 -■ 

(F 4 L) o^ etc. + (l...n)x» . (F X L) + (FJ,)-MF,L)a; 
2.3.4. l...w 



—19— 



-MF 2 L)a + (F 3 L)^ + (F 3 L)^etc. 

2 2 

When (x) is at limit (a), then (#-a)=0, and the series 
above representing (F#)=zero, x=-\-a. Therefore, if we 
change the sign of (a) and substitute it for (x) in the 
series, the series is zero. This is proven by the double, 
identical, derived series, which, if (x) be minus, is iden- 
tically zero, and F#=0. 
x* \a ab \abc\ 

+ |6 x 3 + ac x 2 \abd\x. 
\c .... be -\-\aed\ 
\d\ \bd\ \bcd\ #=— a. 
ad\ 
cd\ 
a 4 — a 4 — a 3 b +a 3 6 -\-a 2 bc — a 2 bc 
— a 3 e -\-a 3 c -\-a 2 bd — a 2 db 
— a 3 d -\-a 3 d -\-a 2 cd — a 2 cd 

— abed = — L^+L'^O. 
This presents the two identical parts into which the 
(Fx) is resolved by the series of derived functions abov'e. 

(Fx) consists of two identical parts when (a) is sub- 
stituted, or, if (b) be substituted or any limit of (x). 

Series (1) (p. 8) is a double series. A series when (x) 
is the distance between the limits of (y) ; a series when 
(y) is distance between the limits of (x). The double 
nature of (Fx) and series (1) (p. 8) is not the same. 
(Fx) includes two identical parts. Series (1) which con- 
tains all the limits of (Fx) is two identical separate series, 
a series for each variable. 

When a, b, c, d, etc., each stand for a definite number 
of units, then the variable coefficients become constants 
and (Fx) has (n) factors, bionmials. If (x) = (a), (b), 
or (c), or (d), etc., then (Fx) is a numerical equation of 
the (wth) degree==zero. 

We will now prove that (Fx)=x n +A 1 x n - ] -\-B'x n - 2 . . . L' 
can be resolved into (n) binomial factors, (#-f-a), (x-\-b), 
etc. 

This we assumed or stated at the foot of p. 18, and 
pn p. 17 hypothesized a series, (x n -\-A l x n - 1J f-B 1 x n - 2 . . .L) 
which was the product of (n) bionomial factors, which 
we could do, but this is an hypothesis. We now want to 
show that x n -\-A x # n_1 -fi? 1 x n ~ 2 . . . . L=(Fx) is composed 
cf (n) bionomial factors, when (L 1 ) = (^) 1 . 

When (x) is constant the successive derived functions 
are: 

—20— 



^ {F 1 A')x^-}-(F 1 B l )x n - 2 -i-(F { C')x^ .... (F,L). Aid, 
with (x) variable, waj n - 1 +(n-l)A l x n ~ 2 -\-(n-2)B l x n - 3 .... 
These two are the differentials of (Fx). Integrate both 
with (x) variable, and the results will be: 
(F.A 1 ) x« +CF.BI x n -l +(F 1 C>) x»- 2 .... (F.L 1 ) x. 

n n-1 n-2 

a^+A^-l+B'tf^+C'cc 11 - 3 .... 

This last series is the original function. The series 
above it must also be the same function. 
(FA 1 )^ !. (FB')= A ] . ( FC) ^B 1 (FL')= K ! . 

( n ) (n-1) (n-2) ( 1 ) 

(FKW). (FJ , =2I 1 ). (FHI=3G') ... (FCW8B 1 ). 
(FB^OA 1 ) • (FA=10) = (n) . 

This relation of (A^'C 1 ... Li) in (Fx) is that of suc- 
cessive derived functions. (L 1 ) may be divided into (n) 
factors. These factors will determine the relative 
magnitude or numerical power of A 1 , B 1 , C 1 , etc. 
But (a, b, c,) etc., are variables, and each may have an 
infinite number of values so that these values will adjust 
themselves to each other so that there will be (n) numer- 
als, so taken, that (x-\-a) (x-\-b) etc.=the numeral (Fx). 
This is what is proven on (pp. 20-21). If (a±\/b) be a 
limit of (x), or a part of a binomial factor as in (x) ± 
(a±\/b), then, in differentiating L 1 , (a±V&) is a unit of 
measure. Its differential is (+!)• When we have in the 
third degree function [x J r (a-\/b)][x-\-(a+yb)~] (c) =x 3 
-f(2a)tf 2 +(2a). 
(c) (a 2 -b) x+(a 2 -b)c. 

(a 2 — b)c=(a-\-\/b) (a — \/b)c. If we differentiate this, 
we have [(a+y&) + (o-* Vb)]c+ (a— y/b) (a+V b) — (2ac) 
-f (a 2 — 6)=coefficient of (x) . (2ac) + (a 2 — b) becomes 
(2a) + (2c) + (2a). This divided by (2), the factorial, gives 
(2a) + (c), the coefficient of (x 2 ) . 2a+c gives (3) ; divide 
this by its factorial and we have (1), the coefficient of 
(x 3 ). 

This illustrates the application of the law of derived 
functions as to the coefficients of (Fx) . 

SOLUTION OF EQUATION OF THE THIRD DEGREE. 

Substitute (a+y&) for (x) in the function of the third 
degree. 

(a+V&) 3 +A(aH v / &) 2 +B(a+v / &)+0=F(a+V&) or (Fx) 
[3(a+V&) 2 +2A(a-fV&)+B]i/ ^F^a+y/b) (F n _ lV ). 

—21— 



[6(a-% 6)+2A] y*/2 =P 2 (fl+y6) (F n _,y). 

(6) t =F s '(a+v'6)(F lua V). 

2.3 

Make (+\ 6)=zero, divide by factorials, and make (y) 
=(+VWj an( i rewrite the above, and we have: 

a 3 -f-Aa 2 -fBa+C 

(3a 2 -f-2Aa-fB)v& 
(3a+A)& 

(+DV& 3 

If Fx=zero, then the sum of the rational terms will be 
zero. 

And A-hAiH-Ba-|-3a6-f-A&+C=0. 6=— B— 2Aa— 3a-. 
This is obtained from ( 2ac ) 2a) 

)=Band )=A. 
+ (a-—b) +c) 

Substitute this value of (b) in the equation and we have: 
— 8a 3 — 8Aa 2 — 2(A 2 -}-B)a— AB— C=0. This will contain 
one integral root which in the numeral coefficient is a 
AB— C 

factor of when the equation is transformed 

8 
into one with integral coefficients. The infinitesimal ele- 
ment of the root is eliminated by the above equation and 
(a) of (a-}-\ b) is a root. Let us solve the equation. 
8^ 3 _26x--fll^-r-10=0. x 3 — 26^-1-11 x— 10^=0. x=^y. 

8 8 8 8 

Transform and y z 26 y z 11 10 

8 3 8 3 8- 8 

1/ 3_26 1/ -^- r 88i/-f-640=0. 

_8a 3 — 8Aa 2 — 2 ( A^B) a— AB— C=0. A=— 26, B=88, 
C=640. 

—8a 3 +208a 2 — 1528a— 2928 =0. Divide by —(8). 
a 3 — 26a 2 —191a — 366=0. 366 contains (a). 

366=2.3.61. 
(3) is a root of the transformed equation, a=3 8. A in the 
coefficient is (— ). (— a) = (— 3 8). —2a— c=A, 

26 6 5 
C= = . Divide bv x — 5 2. (C) is in the 

8 8 2 

coefficient. 
Sx s — 26jt=— IIj-— 10 x— 5 2 



lQx*—o2x-—22x+20 2x—5 



Sx 2 — 6x — 4 



.99. 



> 6 6 9 4 9 41 

x 2 x=4/8. x 2 x-\ = 1 = — 

8 8 64 8 64 64 

/41 1 

±V— = ±— V41. x— -1/8 (3±y41). 
64 8 

We have introduced this example to show that in its 
beginning we used our differential method without (Dx) , 
and multiplied by the use of series (1), (p. 8). Series 1 
=zero. It contains all the limits of (Fx). These limits 
may be developed by it. 

The following is a similar formula for the elimination of 
(±yfr) from the equation of the fourth degree: 
— 8a 4 — 8Aa 3 — 2 ( A 2 +B) a 2 — ( AB— C) a+D=0. 

I have not gone higher than the fourth degree with this 
method of solution of equations. But it seems to me that 
a general solution may be found by eliminating the incom- 
mensurable element. 

FERMAT'S THEOREM. 

(x n -\-y n ) is not =(z n ) when (n) (2). 
Series (1), (p. 8), gives all the limits of (x' 1 ) when (y) 
equals the distance between the limits of (x). (x n ) has 
all its limits in s e ries (1), (p. 8). 

Series (1). —Fx+iF^) (F n _ x y) + (F 2 x) {F n _ 2 y) + {F,x) 
(F n3 y) .... +Fy=0=x n +Nx n - 1 y+N(N— l)x«~ 2 y 2 /2+ 
N(N— 1) (N— 2)z n - 3 y +N(N— 1) N— (N— l)y° 

273 1 N 

Series (1).—Fy+ (F x y) (F^ + ^y) (F n _ 2 x) + (F,y) 
{F nz x) .... +Fz=0=2/ n +Ni/ n - 1 #+N(N— l)y n ~ 2 x 2 /2+ 
N(N— 1) (N— 2)r~ 3 £!_. . • +N(N— 1) . . . N — (N— l)x * 

2.3 1 .... N 

. The two series, marked (1), are each series (1), (p. 8), 
and identical- They combine all the successive derived 
functions of (x) and (y) in each series. These functions 
derived are paired in the same way. Every two has a like 
two in the other series. These series are equal to zero, 
which has been proven both logically and by problems 
solved on this hypothesis. All the work done in these 
pages has been the result of this hypothesis. The logic is 
clear. The work following, the demonstration and proof. 
Then they are equal each to the other- 

—23— 



Now the series c — ^.r-- 1 y . . . — y n ) and the series 
(fP+ttU* -1 ^ • • • — e") are contained, each, in series (1). 
They are identical and equal- They contain all the pos- 
sible limits of (Fx) and (Fy). (jc") is a limit of (F:c). 
) is a limit of (Fy). 

In the series (1) these series U'~ — ncc-^-y etc.). and 
: -~-ny-- l j: etc.) approach infinitesimal zero. At zero. 
they are equal to zero, and are equal to each other. 

Let Z-=ij-—ny n - 1 x—n{n — l)y~- : x . . . — x ,zl . 

~Z 

Then 1 =. — —y—nin — l)x'' 2 y . . . — i 

2 

Then :.-— /-=2z~. - )={z "V 2 )- 

The sum is therefore not=: r -. 

This proof has two series, and the sum is (2s") 

In the second degree, series (1) gives x s -\-2xy-\-y\ Dif- 
ferentiate, and 2: — 2;:— 2> — 2y. 
Integrate, and we cave — -— : — >/-. 

This is not the form of a second power, we may inte- 
grate so as to give back i .-— 2;;?/— y-) . But these two 
forms of integral result show that the second degree may 
give 2.z- or it may be z~. In the third degree we have: 
..'— Sx-y— Sxy-— ir. Differentiate and. — 
3./-— 6j-y— '3j;-— 6;< y-\- 3_ ---oy'-. Integrate, and. 
-'- - — o;ry-— % — 3..- ;-~3./:<- - ... — if. 

This gives the series twice, once for each variable, {x) 
and (y). We always have 2z~- for (Fy) and (Fx) when 
n> 2 

If we take the general series (1). F;<:—F. x(F m .y)—Fy= 
second degree. Series (1) (*>-}- (Fjc) \t . .^ - I : • (F&) 
—Fai. Integrate this. (Fjx is constant.), (F-jj is constant) 
and we have: — F;>:—CFy—DF:>'—Fy. This is a true in- 
tegral of series ( 1 ) . second degree. We could have re- 
stored the form in series 1 1 1 . yet here is another -form 
that expresses the values for the function of the second 
degree. 

In the second degree there are two limits when (Fx)=Q, 
so (y) must be constant, and there will not be two series 
for (.'-—,/-). and 2z- falls out. This proof, therefore, does 
not include the second d e gree. If we take series ( 1 ) , (p. B 
ror the third degree. F:r- \F.z) (F^y) +(F*x) (F- y)-Fy. 
Differentiate, and F^-f (F t x) (F»y) -f (Fjcj (F, y)-iF 

-24—- 



> In the third degree (n — 2) = (1). n=3. 

Also the third (Fx), or F 3 x=l, or constant, and so of 
(?/), then, F.x+F.x+^.x) (F n _ lV ) + (F 2 x) (F n _ { y) +F lV + 
F x y. 

Integrate andFa;+Fa:+ (F r x) (F n _,y) + (F.,x,(F n _,y) + (F } x) 
(F n -i2/) + (*» (F n _ 2 y)+Fy+Fy. 

Here the integral gives the two series. And we must 
have (2z n ) for (x n -^y n ) above the second degree. 

In the first degree (y) does not appear as there is but 
one limit of (x). There is not distance between limits. 
(a* — y) does not approach (x). If (x n -\-y n ) , when n>2, is 
a perfect (nth) power, then 2z n is a perfect (nth) power. 
But (2z n ) is not a perfect (nth) power. So (x n -\-y u ), 
when w>2, is not a perfect (wth) power. 

The differential law, (±na ,nl , ±^ n_1 ), is derived from 
variable factors, not equal to each other. Thus: a, b, c, d, 
etc. are idependent variables, each measured by the same 
concrete unit. In (abed), only one variable can be concrete 
at the same time. Let this be, first, (d) . Divide (abed) 
by (d) . The quotient will be (abed). Let the unit of (d) 

V~dT 
become infinitesimal. The unit of (d) and the quotient of 
(d-^-d) are the same numerically. But the unit, concrete, 
is zero. The ratio of (d) to the function (abed) is (abc) , 
for any values numerical of (a, b, c, d) . (abc) is the 
rate of change when the function and its measure (d) are 
at zero. The function reaches the limit zero with its vari- 
able measure (d) . The increment is the function. Now 
if (c) be the unit of measure, we have for the differential 
(abd) ; with (b) , we have (acd) ; with (a), we have (bed). 
The sum of these rates of change from zero is abc-{-abd-\- 
aed+bed. This is the total differential for (abed). If these 
factors be made equal, then (abcd)=x i , and we have 4x H 
as the result. The differential of (d) is (d°) , or J^d), and 

(d) 

is always ( + ). If we have ( — d) , then (— -cO==(-fl). 

These factors may be ±a, =p&, ±c. The factors will retain 
their signs as multipliers. 

When the differential law is derived from the series 
(x±y) n =x n ±nx n - 1 y+n(n — l)x n : 2 y 2 /2, etc., then a limit is 
implied. Each member is the limit of the other- The 
limit above when (a;±^/)=numerical zero is zero. It is zero 
when the concrete unit is inappreciable, or zero. 

—25— 



If (x — y) be a limit of (x), and the unit become infini- 
tesimal, then as concretes {x) and (y) are equals. Now if 
we add the hypothesis that *=!f, and one of them is neg- 
ative; then, with these two hypotheses combined (x — y), 
(both at zero), (y) and (x) represent the same variable, 
and if {x) be made=0, then function (x — y) becomes 
function (x) . 

The series F(x — y) is zero on two hypotheses numerical, 
and infinitesimal, or, concrete. Every series derived on 
these hypotheses is zero and contains every limit of (Fx). 
Also, the law ( —nx-- 1 ) or ( — ny-- 1 ) is derived from the infin- 
itesimal hypothesis ; and, so any series formed of succes- 
sive derived functions is zero. Differential (ar)=dif. (y) 
for (x — ]/) = (]/ — x). The differential is minus or (+) in 
this series. 

On page (25) the differential (d), or (c) , etc. was ( — ) 
always. But above, (y) , the differential of (-t~x) is 
( — ). In F {abed) , (p. 25) > the differential of infinitesimal 
is absolute. Distance between (a) and (b) is not indicated. 
{x — y) is both infinitesimal and numerical by hypothesis, 
(x) and (y) are negative to each other. (Fx) may or 
may not be = zero. F(x — y) is always zero. A neg- 
ative differential makes (Fa*)=zero. When (a) or (b) is a 
definite limit of (x), then (a*=a), and (x — a)=0 and 
(Fx)=zero. And then dif. (a) is negative to (x). 

VARIABLE COEFFICIENTS. 
.,^+Ax n - 1 +Ba: n - 2 +Cx n _3 m —L=(Fx). 

When this (Fx) represents a line or magnitude, A, B, C, 
etc., to (L) (not included), are constant, and vary only 
within the limits they embrace. These limits are constantly 
changing as (L) varies. 

Fx=x a -\-Ax n - 1 +&x n - 2 +Cx n - 3 +I)x n -* .... =hZ. 
Fy=y n -i r A 1 y 1 -- 1 ^ r B i/V+C i/^+D f- 4 . . . . Z . 

F.7-=x n -LAa* n - 1 +Ba- n - 2 +Ca* n - 3 +Da* n -* .... ±Z. 

F 1 x-^F 1 x^r(F 2 x) (F_ x y) + (F 2 x) (F_aj)-(F z x) (F- 2 y) + 

(F x) (F_ y) etc Identical zero when limit (x — y) 

approaches limit (x) , and (—wo* 11 - 1 ), or (^zWi/ 11 - 1 ), are the 
laws of derived functions. 

In series (1), when (Fx) and (Fy) as above are sub- 
stituted, there will be variables in the series. (A 1 ) 
multiplied by its factorial into the last coefficient of (Fx) 
and factorial, — (B 1 ) and factorial into the next to the 

—26— 



last coefficient of (Fx) with factorial, and so on, forming 
a series. Using the fourth degree to illustrate. 

(z 4 +A:r :! +B;r+Ca;+D) (24) = (Fx) (F 4 y) . — F 4 t/=24. 
(4^ 3 +3Ao; 2 4-2Bx*+C) (6A 1 +240)"= (*» (F^y) 
(12^ 2 +6Aa;+2B) (2B 1 +6A 1 y+12y 2 ) = (F 2 x) (F_ 2 y) 

(24a+6A) (C 1 +2B 1 i/+3A'2/ 2 +42/ 3 ) = (F 3 ^) (^-stf). 

(24) (D^C^+B^+Af+rt-^ai) (^_ 4 2/). F_ 4 #=24. 

24D+6A 1 C+4BB 1 +6AC +24D', are the variable fac- 
tors =D(F 1 D 1 ) + (^D) (F 3 D 1 ) + (F 2 D) (F 2 D') + (F 3 D) 
(F 1 D 1 ) + (F 4 D) (D 1 )=0, by the law of series (1). 

A, B, C, D, etc., have each (n) general limits of (x). so 
that (x — a), (x — b) , approach (a) and (b), and these 
coefficients are so related that they form a series of de- 
rived functions. A 1 , B 1 , C 1 , are limits of (y) , so that 
(y — a 1 ), (y — b 1 ), approach (a 1 ) and (b { ) just as (x — a) 
approaches (a) as a limit* of (x). It will be seen in the 
illustration just above that when the variable 
terms are left out of series (1), that the remainder of 
series (1) may be so combined as to constitute different 
powers of (x-\-y). But (x-\-y) is zero by hypothesis, and 
so the series (1) is zero when the variable terms are added 
as shown above. And series (1) is zero as shown by the 
first derived function of the above series. 

(x) includes (a), (b) , (c), etc., the (n) general limits 
of (Fx). So (y) includes (a 1 ), (b 1 ) , (c'), etc., (n) gen- 
eral limits of (Fy) . (a — a 1 ) is (x — y) . This will make 
(a — a l )=0 when (a — a 1 ) reaches the limit (a), or 
(c — c 1 ), the limit (c). We see that the series of coeffi- 
cients in series (1), above, becomes zero on the same 
hypothesis that series (1), (p. 8) is zero, (a) becomes 
(a 1 ) by the infinitesimal unit; also, by (a — a 1 ) approach- 
ing (a). (a — a l ) = (a) when (a)=0, and (a 1 ), the dif- 
ferential becomes (a). In (Fy) = (y n -\-A 1 y n - i -\-8 l y n - 2 . . . 
-fZ 1 ), the values of (y) are grouped by a general law of 
relation to each other, a 1 , b l , c 1 , etc., but related to! each 
other by the derivative law of coefficients. As (a 1 ), (6 1 ), 
(c 1 ), etc., are differences between (a), (b) , (c) , etc., (a) 
has the same relation to (a 1 ) as (x) to (y) ; only, (x — y) 
approaches any value of (x) represented by (x) , for (x) 
is the general limit, but (a — a 1 ) approaches (a) or only 
one of the (n) general limits of the functional law, (Fx). 
If Fy=0, then only (n) limits, fixed limits, can be given 
to (y) . If we then give a fixed value to (x) in (Fx), 

— 27— 



then (F -~ould have (n) values for (x) corresponding 
:: the (n) values of (y) . And Fx=0. But if F =. 
we are at liberty to choose only one value of " :. 

values are fixed by iF\. =zero. Also, if we make (Fx =-- 
zero an :'. A I \ ' -: : . constants or numerals, then only one 
v-alue of (y) will be given in series (1). This value of 
(jr) will give the corresponding value of and for th^se 

values ::' . and (y) the {Fi and F% equal zero, and 
:;.t series 1 =■] 

(n — 1 values :: if) are determinant- :: 

THE DIFFERENTIAL LAW +NX" AND -XY-- . 

A Sum of Vi i E ' 

The differential law, ±«r J J or — ny a - , or, sum of vari- 
able ration is the law of combination of things taken 

(n — 1) together. If we consider the factors of as 

all different, we shall hav- ; | i -:; . to (n factors. 
?.:: =: Then abed:= vsrhen the factors are equal 

The function (abede) gives five ratios, one ratio to each 
variable elemen: Now if '-: be supposed to be at 

infinites imal zero, the ratio will be a ratio of the variation 
:■:.:. :'.:- sum will be abed — abee — abde — aede — be:'-; 

d t b 

but a, ':_ and so on would be unity and would not affect 

b 
tie above quotients, [i ; is differential of (&)]. rhe 

:■- : : ~- : : the law of coeficients as we have already shown 
in other pages in the discussion of this subject. If we 
put (abede )=x% the above variable ratio is Sac 5 -*, which 
is the law I - - - ■ when ( n = " 

The above ratio is taken at zero when the unit of meas- 
ure is (1), the differential of (a- r": and we 
havr different combinations of the elements of the 
function, each divided by an element whose unit Ls infini- 
tesimal. This contains all possible combinations of the 
elements of the function, and so is the total of its ..;.nge 
or variable power. It is therefore the universal value of 
magnitude as it measures it from its initiative or zero. 
AD forms of expression of magnitude when affected by the 
abc t ratio reach a limit at zero- 

The Foove is therefore the general proof of the differ- 
ential lav - When the differential is a distance 



between limits, thenjt ,n = ±nx n ~ x , ( + ) when ( — y), ( — ) 

y 

when (-\-y), as has been discussed in previous pages. If 
we take Fit , =-a; n 4-Aa: n - 1 +Bx ,n - 2 , etc., and begin to differen- 
tiate from the right with (x) constant, we have dif. C= 
(n—2)B, dif. B = (n— 1)A. Dif. A=n, dif. x n =0, 
and naP^'-fCw — l)Aa; n - 1 +(n — 2)Ba; n - 2 etc. The differential 
or derived function of each term is the same with either 
element constant. Again, if we take (Cx°) and integrate 
successively we have Cff°4-Bff-|-2Aff 2 -|-2 . 3x*- \-0. In this 

2 2.3 

we took the successive derived functions of (C) and inte- 
grated (x») or C-{-(F 1 C)(F n _ 1 x)-\-(F 2 C)(F^ 2 x) + (F s C) 
(Fn-3%) -fO=series (1). Any term will give by this law 
the rest of the Function. If we put the series of coeffi- 
cients on a preceding page into series substituting (x) 
for (a) and (y) for (a 1 ), which values (a) and (a 1 ) stand 
for in the series, we have for the fourth degree, [(#) = 
(a) or (b) or (c)], 24 D+6A 1 C+4B'B+6AC+24D. 

24x* 4 +6.4.4ic 3 2/+4.6.6a; 2 ?/ 2 +6.4.4^i/ 2 +24z/ 3 . 

x 4 -\-4x s y-{-6x 2 y 2 -\-4xy 2 ^ r y 3 . 

Rut (x) includes (a) and (a 1 ) is (y) , and when (x — y) 
=0, (a — a 1 )=0, or, otherwise expressed, (a) is a group 
of values of (x) and (a 1 ) is a group of values for (y) . 
And the series of coefficients of (a: ) like the rest of series 
(1) is divided by (x±y) and the series is zero when 
(x — y) or a — a'=0. 

Tf (Fx) be the limits of a line, then (Lx°) will represent 
both a variable unit of measure=(;r ), and a variable 
position of measure on the axis of ordinates, variable 
(L). If (L) is constant, the origin is fixed. When abso- 
lute zero changes position on the axis of abscissas, then 
the coefficients of (x) for the higher powers, or powers 
above zero, are variable. When the latitude and longitude 
of absolute zero is variable, that is, at any point within 
the magnitude, then the coefficients of all powers of (x) 
are variable, and (Fx) is a general statement of limits- 
This will explain what may seem to be confusing in these 
preceding studies. This matter is more fully illustrated 
in connection with the line further on. 

In the discussion from page 26 to 29, inclusive of these 
pages, A, B, C, D, etc., are called constants, when zero is 
a fixed and not a variable origin. In the (Fx) , (A) = 
a-j-b-\-c, etc., limits of (x) , when (x) is supposed to equal 
a fixed number of units. Then (Fx)=0. But (a) is a 
variable term and may have any value. When (Fx) =x n -\- 

ro 



~"r./. : ir.star.: ~e hav-= 



- — - efer is variable then . > are variables, 

- ~ - - : A. B. C. etc.. are derived functions 

=(K), the last --rm. of the function. It is in 

t are used as constants, bat variables in 

n to each other, bnt= 26 they are treated 

ustants This is not rrue to series (1 as :-.'.'. its ele- 

t t variable except when for a numerical limit, 

F =iri then series 1 =zero for fixed values of (. 

Let us develop F witb :his hypothesis 
-{-A — Z — Z - A Z 7 : _:, we have, 

-_--. -B 

-2A 

— A^— E - 

: -:.-.. -B 

; -2A 

I: - -- - : then (cf)=0, and (D)=0. 

But re differential of (T>=abcd) is (abc= We hen 

- 

+A — I -C 4S =<Fj)(F,#) 

-— iaj -: .. -— = - 

24 - SA — 4A -1Z =7 ~ 

is (r— - 1 -'- — B 1 -— C)= ~. p»_ 

Pat =0, and = and divide by 16). A"-=A. 

et: 

-.-- -z - : s 

! -2A--B.A =.~ 

I -A Z ::."=■: :her. 

1)C ;.-. -- ---. --.I -2AB-3C=0. 

No magnitude nor any function of any number of vari- 
ables can have more than one independent variable. Let 
F I represent general ; function of variables. 

Let us suppose and e :: have :ach an inde- 

t t iiirerer-tia] :r limit Then, Fix, y, :J = 

F ; -\-F : —F : : N :~. if we inte- 

t these we have F : Huree times which is not 

r original F . : 

Now, if we make the differential and the in- 

t v . i : ar ■-: fur. et: ens :f 

terms e containing (r) and are by hypo- 

11 be zero; for, (<f, applied to them, 

: era Only one term will remain, and we have, as inte- 

original Fix. y. z). This would be so if (y) or 

— SO— 



(z) were assumed to be the only independent variable. 
This proves that x n -\-y n is not=(2 n ). (z n ) cannot have 
two limits at once. Therefore the hypothesis that (x) , 
(y), and (z) have an independent limit each in the func- 
tion cannot be true; for, the only function they produce is 
not F(x, y, z,), but this function is reproduced on the 
hypothesis that only one of the three is the measure or 
independent variable. To illustrate, — (xyz), with three 
independent variables, gives, for the differential, (xy)dz-\- 
(xz)dy-\-(yz)dx. Integrate and we have Sxyz=xyz, which 
is not true. 

If (x) is independent variable, then we have (xy-\-xz-\- 
yz)dx. The terms in the parenthesis are by hypothesis 
constants, (dx) is zero for the terms having (x), and 
the integral is (xyz). If we suppose (z) or (y) to be the 
independent variable we should have the same result. 

We have used (dx) in the above proofs instead of the 
limit (y). This is done to conform to the terms used in 
the ordinary theory, (dx) is the limit of (x) and is equal 
to (x). (y)^=dx as a limit in the discussion of this paper. 

GEOMETRICAL ILLUSTRATION. 



1 

1 




a 




r 
-i 






p 








a' 
c 






P' 





Let abed be any rectangle, ab=x, bd=y. Also, ultimate 
hypothesis (p. 5), a&=12, frd=8 . (bi) =di£. (y)> (bp) = 
dif. x. The figure (a 1 b 1 d J c ) will represent (dx) and (dy) 
approaching (x) and (y) as the figure and its unit ap- 
proaches (c) and zero, when the differentials will=the 
variables, and (x) must be put=(?/) and y=(x). Only 
one variable can exist and that is the unit of measure ; and, 
(dx)=l y and (dy)=l, and only one of the variables can 
be independent. But take the above figure, (abdc) . Put 
dif. (x)=2, dif. (y)=l. 

#i/=rectangle. Dif. (xy)=xdy-{- ydx=28. #=12. dy=l. 
(x)dy=12. (y)dx=lQ. But this is not the differential of 
the rectangle above, but of another figure twice as large, 
and which has this rate of increase. 

—31— 



Integrate the above and we have 2xy= 192= the rectan- 
gle whose proportions agree with the given differentials 
above. 8 and 12 are not the sides of the rectangle, but 
\/96 is one side and 2\/96. 2y96x y 96=192; and the 
rectangle agrees with its variables so long as the differen- 
tials retain the ratio of (2) to (1), which makes one vari- 
able dependent on the other. With one independent vari- 
able, (y)dx integrated=:n/. The rectangle has (ax) and 
(bx) for its sides, or (x) and (y) when (y) is a limit of 
(x) ; or, (x) is a limit of (y) . 

A magnitude has one concrete unit of measure, and only 
one collective unit or number, to which all other numbers 
must bear a fixed relation and so become functions of this 
variable number. All functions become zero on the hypo- 
thesis that the unit" or differential approaches infinitesimal 
zero. 

The general differentials of lines, surfaces and solids are 
taken at the limit zero. The magnitude exists in all its 
proportions as an infinitesimal, and always has those pro- 
portions, (x) is a general limit, (y) is the distance to a 
limit from which (x) is approached, (x — y) is the dis- 
tance from a point of measure to the last limit named 
above. Put (x — y) at the limit of the infinitesimal unit, 
and both reach the point of origin or measure. Put 
(x)=0, or absolute zero, numerically, and (y) is at the 
first limit (x), and will be so as the unit is restored to its 
original size. Then (Fx), which was infinitesimal zero, 
will be restored. The limit (x) is fixed and \Fx) also. 

If we use the above law of integration of variables in 
restoring the coefficients of Fx-=x nJ rAx n ~ 1 -i-Bx n - 2 , etc., we 
have as coeoefficient (x n )=n. Put w=4. Then there are 
four units and so four differentials and four variables, 
(a+6+c+cZ)=A. In the above (Fx), the series of derived 
functions runs as follows: F,C=B. F 1 B=2A. F 1 2A=3. 
F 1 D=C. F 1 C=2B. F.B— 3.2A. F,3.2A=4.3.2.1. In in- 
tegrating above we follow the reverse order of these numer- 
als, (a+6+c+cZ)-^l=A. 

"We then integrate (a+fr+c+cZ), making each an inde- 
pendent variable, and we have, for (a), (ab+ac+ad) ; for 
(b), (ab+bc)+bd; for (c), (ac+bc+cd) ; for (d) , 
(ad {-cd+bd) . Added, the sum is 2(ab-\-ae-\-ad-*-bc+-.bd 
-\-cd). This is twice the integral (B). If we integrate 
(B) as above we shall have (3C). If we integrate (C), 
we have (4D). We see these integrals do not in any case 
restore the function without division. It will be seen, 

—32— 



however, that this law of integration follows that of the 
variable (x) . (1) integrated gives x, Int. x=x 2 . lnt.x 2 =^x^, 

1 "2 3 

etc. 

# 4 -f Ax*-\-Bx--i~Cx-\- D becomes, when we integrate (x) 
from the right and the coefficients from the left, 
x 4 -{-Ax s -\-2Bx 2 -\-SCx-j-4T>. This does not restore the func- 
4 3 2 1 

tion in form. (a-j-b-fc-j-cO, or (A), will not restore (B) 
its integral when only one variable is used. (F x x) will 
always give (Fx) when only one variable is independent. 
(abc-\-abd-\-bcd-\-acd) , with (a) independent, gives only 
(abed). With (b) independent gives only (abed) , and so 
with each of the variables when the rest are constant. 
(a+b) is the (F x x) when (Fx) is the second degree. 
(a-\-b-\-e) is (F 2 x) for the third degree. (a-\-b-\-c^-d) is 
(F A x) for the fourth degree. (ab-\-ac-\-bc) is (F x x) for 
the third degree. abc-\-abd-\-bcd-\-acd=(F\x) for the 
fourth degree. Its integral when (d) is independent va- 
riable is (abed) ; when (c) is independent variable is 
(abed). When each variable is independent is k(abcd) 
as has been shown before. Divide by (4) and we have the 
integral, (abc) , etc., are third degree functions. Adding 
a variable factor raises it to the fourth degree, and so we 
divide by (4). Int. oj s =*#*/4. (x 8 ) is integrated with one 
independent variable. With four independent variables 
we have (# 4 ) which is four times the integral. In (Fx), 
all the variables used are limits of the one independent 
variable (x) . (a-\-b-\-c) cannot be (F X L). 

(±nx nl ), the law of the first derived function of (x) 
is on the hypothesis that each factor of (x n ) is an inde- 
pendent variable. If we integrate (nx*- 1 ) with one vari- 
able independent, (dx) , or (x) , we have (n — l)^"- 1 con- 
stants; and so have x n ~ 1 Xx=x n . This shows that (Fx, y, 
etc.) can have but one independent variable. In the above 
we must distinguish between variables and independent 
variables. (Fx) has many variables, but one independent 
variable. 

MEASURING THE ARC OF THE ELLIPSE. 

Differential of an arc of the ellipse=rfz= = V | A 2 — ex 2 \ ,when 

VI A 2 — x 2 | 
e *=A 2 — B 2 . 

If (A)=half the major axis and is the unit of measure, 

—33— 



thai (dz)= \ 1 — €-: - 
1 — x- 
Put (l_^i= Put (1— € 2 ar 2 ) 12 =ti. Then, inte- 

grate (Mr) by series (1), p. (8), and we have: 

)y-\-{udr-i-vdu) y 2 -J- {udm s ^-vdFu-j r 2du dv) /_. 

.T2 1...3 

-f- {wPv-\-vdhir^Sd*M dv^-Sdv dru) 

_ , ud v — vd*Ur±4&u dv-\-4d*v du-\-%d?u d*v) V s . 



...5 
-f (Mdrv-\-vd*Ur^-5d*udv-§-5d 4 v du-\-10 d*v dHi+KkPu d*v) 

. . .6 

+ (vd*v-\-vdhtr^-6&u dv-Jr&Fv du±15d*u d^v^lhd^v dhi 

+2(kf a tt dH?) j/_ 

. .7 

— | ud 7 v+vd i w-{-7&u <ft?-f-7d*c du-^21 <Pu d-v-$-21<Fv tf 2 !* 
— 35c?*if d*F-f-35d*t!> d*u)_jf_ . 

...8 
4- (i«i^H}-r<i s it+8<f 7 i* di?-f-8<rr du^-2&d*u d z r 4-28 d*r d 2 !* 
HhS&i 5 !* <IH?-f-56<Pi? d*u+70d*v d*u) y_ 

.. .9 

— < ud^-i-vd^UrJr9dht dv -j-9tPv du -fSM 3 !* d?v f 36 cf 7 ^ d-u 
+84d*tt d*t?-f-84<r*r d^Hf-^Gd*!! d*r+126 d»r <f*M. ) y 1 * etc. 

...10 

In the above series, in each parenthesis, the indices 
proceed to the right in pairs, increasing and decreasing 
by unity respectively, until the indices differ by unity, 
when the series ends, except in the third, fifth, seventh, 
ninth parentheses, which end with equal indices, this term 
being formed by adding the last pair of the preceding 
parenthesis. 

This is the law of the indices. 

The coefficients in each parenthesis are found by adding 
one coefficient of each pair of coefficients to a coefficient 
of the following pair in the preceding parenthesis. 

This is the law of the coefficients, and the series is 
easily extended to any desired limit. 

Now in the use of this series it is necessary to find the 

constants, when, {*) is made equal to zero. This is done 

by obtaining the successive derived functions of (m) and 

I separately and putting (j*)=0. All the constants in 

— 3i- 



> ■ 

the series of (uv) will thus be found, and by substituting 
these in their places in the series (uv), and making (y) 
= (x), we have the integral of (uv) . 

We may develop (u) and (y) each into a series by the 
binomial theorem, and then easily apply series (1), p. (8), 
to each term, and the last derived function will, in each 
case, be a constant of (v) or (u) in the series (uv). 

These constants can be obtained by applying series (1), 
p. (8), directly to (u) and (v) . The terms of each de- 
rived function of (u) and (v) that contain (x) as a factor 
will disappear and the result will be as above', but more 
tedious and lengthy in obtaining it. 

If we substitute for (u) and (v) the factors of the dif- 
ferential of the arc of the ellipse, all the odd indices will 
contain (x) as a factor and will disappear. The result, 
or sum of the series is, then : 

-\-vd 6 u\ -\-vd & u\ 

uvy 1 -\-vd 2 u\ if -\-vd*u\ if + 15d 2 v d*u\ y 7 +2&d 2 v d*u\ y* 
+d 2 vu\..S-\-6d 2 vd 2 u\...5 -{-I5d±v d 2 u\...l +10d*v d*u\...9 
+d*vu\ +d*vu\ 2Sd 6 v d 2 u\ 

d Q vu\ etc. 

r=(l — x 2 )- 1/2 . This is the differential also of an arc of 
the circle. On pages (12) and (13)., the integral of 
(v) = (l — x 2 )~ 1/2 is found by series (1), p. (8). On p. (13) 
the constants of the successive derived functions of (v) 
are found. They are v=l. d 2 v=l. d^v=S 2 . d*v=& 2 .5 2 . 
r 7f ^=3 2 .5 2 .7 2 , etc. 

u= ( l—e 2 x 2 ) 1/2 = ( 1 — e^xj — e*x 4 — Se 4 x\ e tc. 

2 2.4 2.4.6 
e 2 x 2 , — 2e^^, — e~, — e 2 =d 2 u= — e 2 -{- etc. 

~T ~2~ 

—e'x\ —e*x 3 , — Se 4 x 2 —Se 4 x —Se . .SdHi= — 3e 4 , etc. 

2..4 2 2 

d« u =—3 2 .5e«+ etc. d s u=— 3 2 .5 2 7e 8 + etc. 

The above values being substituted in the formula for 
an arc of the ellipse given above on this page, 
we have, by putting (?/) = (x) [(?/) is not the ordinate (y) , 
it is (y) in series (1), p. (8)] : 
a- -I- (a; 3 — e 2 x 3 ) + (3 2 x 5 — 6e 2 x 5 — 3 4 x'° ) + (3 2 .5 2 a 7 — 15.3 2 e 2 3- 7 

2.3 2.3.4.5 2.3.4.5.6.7 

— W.Se'x 7 — 3 2 .5e 6 a; 7 ) + 3 2 .5 2 .7 2 a-°— 28.3 2 .5 2 e 2 ^'— 70.S 2 .3e±x 9 
—28 . 3 2 . 5e 6 x»— 3 2 . 5 2 7e 8 x\ 

2.3.4.5.6.7.8.9 

—35— 



This series contains the integral of an arc of the circle. 
We can leave out the — terms as expressing any portion 
of an arc of the circle and combine the remaining terms 
and we have: 

Circular arc — (£%r — e-x x - — e-x ' — Se'-x : — e-x~ — e z x' 

T73 4.5 2X5 lATi ~44.7 -iX? 
+ 6e 2 x 9 -f- e 4 jr : — e'x< — bt-x", etc. 

6.6.8 4.6.8 6.6.8 4.6.6.8 

If we make ix) in the circular arc and in the ellipse= 
to the half of the diameter of the circle and also half of 
the major axis of the ellipse, then the above becomes: 
lu — ^ ll- — -J— — _ e * — 3£ : — _£l_— _fl — 5g : 
2 2.3 4.5 2.4.5 4.4.7 4.4.7 4.4.7 6.6.8 

— _e_ — _^ — _of etc.) = - — [ (2629— ) e -— (263= ) e* 

4.6.8 6.6.S 4.6.6.8 2 10080 6720 

— (25— ) c - — (5— )e -] — etc. » 

2016 1152 

p.uv—ur. vdv. 
p.uv — 2p.ur.v — ?/r ; r. 

p_;i y — Spur. V — Op. (IT- v — V.T- r v. 

fjj'.i' — \p-ji.T- v — 6p 2 ur,v — -ip. ui%v — urA\ 

pjjy — bpji r .i' — 10p z ur.v — 10pMr,v—bp l ur i i' — ur,r. 

p J u[' — 6pj/r^v — lDp^r^v—20p^rA' — lbp z i(r i v — 6p l urA i — i/r^v 

P-uv—'p/iW. v— 21p.ur._v — Sbpjrr.v — SbpjurA' — 2 lp 2 ur,v — 

lp. U'l'A' — ur.v. 
pAt v — Sp.u r. '■: — 28]."', u r. v — b6p.u r. v — 7 Op a>. r 4 v — b'op-u r, v — 

'2$p._ur f v—$p. ■ '-'.' — urA'. 
p.uv — 9p.ur. v — SGp.u ' r -i' — 84p^/r ; r — 126p-/* 4 r — 126p 4 r ? r — 

S-ip.jt r. v — 36p : u r. v — 9p. u ta- — u ?\ v. 
p.AJ.v — lOpl'urA'— 4op~Acr 2 v— 120p.ur.v— 210p 6 wr 4 r— 252p : wr 5 r 
—210pM r. v — 12~0p z u r. v—Abp-u r_ v— 1 0p : u r. ? v — u r. , v. 

On the preceding page the formula of successively 
derived functions of (u-v T ) is abbreviated by omitting 
what can be at once supplied. p : =p(p — 1), P : =p(p — 1) 

(p — 2), etc. r_=r, r : =r(r — 1). etc. (p_) require? 

(uJ-'-du-) etc. (7\) requires (v~--dv-). etc. 

The numeral indices in each term are the same as that 
attached to the exponents (p) and (r). 

The law of the indices is that they increase from left 
to right by unity with (r), and decrease from left to right 
by unity with (p). The first (p) in each parenthesis. 

The law of coefficients (numeral), is: Add each coeffi- 
cient to the next, in the preceding parenthesis, and the 



sums, in Order from the first will be coefficients, after the 
first, which is found by adding unity to the first coefficient 
above, (du) and (dv) are not differentiated. This is left 
out as it can be added. The differentials of (u) and (v) 
will be factors of the terms of the series. They will con- 
tain the constants; and, in integration, when (x) is made 
=zero, the entire series except these constants and mul- 
tipliers will disappear. 

Every term of the series which has (x) as a factor 
will have no part in the integral. 

LENGTH OF AN ARC OF THE PARABOLA. 

1 (p 2 -\-y 2 ) i/z is the differential of an arc of the parabola. 

V 
Using series (1), (p. 8), as developed in the series (uv), 
or (uPv T ), observing that the series will not contain 
one of the factors, the . right hand column, (u) omitted, 
will contain the constants of series (1), p. (8). The 
results will be similar to the development of (1 — x 2 ) 1/2 , 
the differential of area of a segment of the circle, on page 
(14). (p) in the parabola which is one-half the parameter 
of the parabola takes the place, in the formula, of (R), 
one-half the diameter of the circle; and the plus sign 
gives signs after the second alternating ( — ) and ( + ). 
(y) is used in the formula for an arc of the parabola, 
because the ordinate is taken to be the unit of measure 
of the function instead of (x) . (dy) is unity. In differ- 
entiating, (x) was made unity first, and (y), including (x) 
afterward. The value of the parameter must be taken 
with y=l, when a value is ascribed to (y). y 2 =2px. y 2 = 
2px in the formula p= 1 . 

2x 

The formula for an arc of a parabola will be: 
V+ _V? — 3r +3 2 .52/^ — 3 2 .5^V, etc. 
2.3p 2 ...5£ 4 ...7p 6 ...9p 6 

The above may be abbreviated by cancellation of like 
factors. The results of the above may be compared with 
the usual logarithmic formula, 

y(p 2 +y:) 1/2 +P_\og. [j/+(p 2 +j/ 2 ) 1/2 ] — (p/2) (log. p) 

2p 2 V 

p= 1 in above. y=>l. 

2x 
To explain above formula as compared with one given 
next. The differential of an arc of a parabola when (x) 

—37— 



is the differential unit, we have : y 2 =2px, ydy=p. dy 2 = 
p\ dx=l. Then, differential of the arc = \/ (l -fp 2 ) = 

y 2 W 2 

y- 1 (p--\-y 2 ) 1 -. When (y) is the differential unit, we have 
y 2 =2px. y=pdx. x is a function of (y). dx=j^ dx 2 = 

V 

p 2 p 

The last differential is the one used in the formula given. 
The formula converges slowly,', but, since the denominators 
ultimately contain all the products of the natural numbers 
in their order, this denominator must be the largest nat- 
ural number, and so, the terms approach zero. The pur- 
pose of this work has not been so much to get a practical 
cormula as to show that the principle in series (1), p. (8).. 
is illustrated as universal in its application, and is an in- 
tegral law. This shows that series (}), p. (8), restores 
all functions when (x) is made = ezro and (y) is made 
— (x) . This shows that the series, at zero, is = to all 
functions. Therefore, series (1), (p. 8), is zero both 
numerically and infinitesimally. The series is made in- 
finitesimal by the differential hypothesis, because the dif- 
ferential is a value infinitesimal and (x) and (y) have at 
the same time an infinitesimal unit, and, as numbers, be- 
come equal, and (y) takes the place of (x) ; and, as its 
differential, restores atonce, both the size of the mit and 
the numerical value of (x) , and so restores (Fx) . 

The formula for the arc of a parabola above, contains 
the constants of [1 p(p 2 +2/ 2 ) 12 ] developed into a differ- 
ential series (1), (p. 8). These are taken from the series 
on a preceding page, developing the successive derived 
functions of (vPv r ), or on another page developing (uv). 

We will now develop Lpip'+f) 1 2 , by McLaurin's The- 
orem, into a series ; and integrate it in the usual way ; 
and, then, by series (1), (p. 8). Adn show that the results 
are identical. This will illustrate the practical trutn of 
series (1), (p. 8), as discussed in this manuscript. 
j^( p 2 +2/ 2)i/2 ==1 _^^2 __ j£ _|_ 3^/6 _ 3.5?/ s etc. 

V 2p 2 2Ap* 2.4.6p* 2.4.6.8p s 

Integral = y-\- if — y'° +3i/ T — 35?/° etc. 
2.3p 2 2.4.5p 4 2.4.6.7p 6 2.4.6.8.9p 8 



-38— 



By series (1), (p. 8) : 
% (1+ y 2 — y* + 3i/ 6 — 3.5.1/ 8 etc. 


) y 


2p 2 2.4p 4 2.4.6p 6 . . 2.4.6.8p 8 

{y — y z + 3r — 352/ 7 ; 


1 2/" 


p' z 2p 4 2Ap & 2.4.6p 8 
( 1 _ 3i/2 + 3>52/ 4 _ 3.5.7^6 ; 


.2 

) y z 


p 2 2p 4 2Ap 6 2.4.6p 8 
(Sy + 3.5?/ 3 — 3.5.7i/ 5 ; 


2.3 

I 1/ 4 


p 4 2p 6 2.4p 8 
(_ 3 + 32.5^2 _ 3.527^4 j 


2.3.4 

1 y 5 ' 


p 4 . 2p 6 . 2.4p 8 
(-\-&5y — SS 2 7y 3 ) 


2.3.4.5 

1 y* 


p* 2p 8 
(3 2 5 — 3 2 5 2 7i/ 2 ] 


2....6. 

1 2/ 7 


p 6 2p 8 

(3 2 5 2 72/) 


, 7 

y* 


p 8 
[—(32527); 

p 8 


...8 
~/9 



Put (2) for (1/) in this right hand column. The use of 
(y) is confusing as it is not the ordinate. 

Note — The Integral at the bottom of the page is the 
same as the one at the top. 

The constants and multipliers: 

y+y* — 3?/ 5 + 3 2 .5j/ 7 — S 2 5 2 7y° etc. 

2.3p 2 2.3.4.5p 4 2.3.4.5.6.7p 6 2.3.4.5.6.7.8.9p 8 

If (x) be taken as the differential unit, the measuring 
unit of the function, then we have {dx 2J rdv 2 ) 1/2 =y 1 
(p 2 +y 2 ) 1/2 . 

y- 1 (p 2 -fj/ 2 ) i/* =pr i-|- y — y* +_37/ 5 

2p 2Ap z 2.4.6p 5 
(— py~ 2 + 1 — jfy^ + 3.5t/ 4 — 
2.4p 3 



2p 

(+2p2T 3 



2.4.6p 5 
% + 3J52/ 3 
4p 3 2.6p 5 
2.3pr 4 — 32/ + 3 2 .52/< 



— 3-5j/ 7 etc.) y 
2.4.6.8p 7 

3.5.72/° etc.) y*_ 
2.4.6.8p 7 .2" 

3.5.72/ 5 e tc. ; y z __ 

...3 

4 



2.4.8p 7 

(_2.3ff2T 4 — 32/ + 3 2 .52/ 2 — 3 .5 2 .72/ 4 etc. ) j/ 

4p 3 2.6p 5 2.4.8p 7 ...4 

(+2.3.4ff2T 5 +3 2 .52/ — 3 .5 2 .72 / 3 etc. ) y^_ 

2.Sp 7 ...5 

-3 2 5 2 72/ 2 etc. ) y^_ 

2.Sp 7 ...6 



6p 5 

(_2.3.4.5p2/r 6 +3 2 I 5- 
6p 5 



-39— 



(-f-2.3.4.5.6pr 7 — 3 2 .5 2 7y etc. ) y^ 

$p 7 ...7 

(+2.3.4.5.6.7pr 8 — 3 2 .5 2 .7) y* 

Sp 7 778 

y 2 — 3# 4 + 3 2 5j/ 6 — 3 2 5 2 7i/ s 

2.2p 2.3.4.4p 3 2.3.4.5.6.6r 2.3.4.5.6.7.8.8p 7 etc. 

In this left hand column the value may be p(l — V2+V3 
— Vi+Vs) e ^ c -> added to the series below, 2=1/, when 1/= 
zero, then (py~ l ) becomes {p). 

py 1 cannot be integrated in the usual way. So th'i 
result cannot be compared with the usual integration by 
series. p|/° is infinite. This is readily met by series one, 


for (py 1 ) becomes zero. In the former series (y) is the 
unit, 2px=l. In this latter series y 2 =2p, y=\/2p. 

Pages 39 and 40 is a stud^ incomplete. 

In series (1), (p. 8), the differential is (y). (y) at 
zero is = to (x) . In this theory the differential is the 
general distance between limits that contain the variable 
and the function. The differential is always = to the 
variable. The differential of the function is the function 
between any limits. The element of time disappears. The 
function is whatever its unit, (concrete), is. 

(y n =ax) is the general parabola. 
. .ny n ~ 1 =adx, when (y) is the unit (dy) . 

ydx=y (ny n - 1 = n i/ n )— differential of area. 
a a 

jt Int. y n =n [y^z+ny*- 1 z 2 -J- n ( n — l)y n ~- z^ z^j 

~a ~a 2~ 23 n+1 

If (n) be an intiger, then, the factorials will end as 
above, (z) will reach the (nth) power with the next to 
the last term of the series; and, the exponent of (y) will 
be unity in the same term. We then integrate (z n ) as re- 
quired in series (1), (p. 8) and we have z"- 1 for the last 

n+1 
term. Now make (y)=zero and put (z) = (y), and the 
series becomes n f/ B+1 . For (i/ n ) substitute (ax) 

a(n+l) 
and n axy= n £2/+C=area of parabola. If (n) 

a(n-\-l) n-\-l 

=2, we have 2 / 3 #2/=common parabola, (z) above is used 

—40— 



insteacLof (y) in the series (1), (p. 8) because (y) repre- 
sents the ordinate of the parabola. 

We use the above expression for the area of the general 
parabola to illustrate the proof of Fefmat's Theorem. 

n [y n+ * + (n+1) y n z-\- (n-\-l) ny*- 1 z 2 -f 

(rc+l)a 2~ 

(n+l)n(n — l)?/ 1 - 2 z 3 -f (n-j-1) (n) (n — 1) (n — 2)y n ~* z* 

2.3 273^1" 

.... + (n-fl)n . . . [n — (n — 1)] z^ i_ 

(n-\-l)n...[n — (n — 1)] (n+1) 
Omitting the factor n , (y n ^)=the sum of the 

(n-j-i)a 
remaining terms of the series, because the series is = to 
zero, and contains all general parabolas. The series is = 
to zero for any values of (y) and (z). When (y)=zero, 
we have (z n+1 ), and (z) may have any value, and so in- 
cludes (y) . The form of the series is that of a perfect 
(nth) power. Now. if (y) is the measure of one para- 
bola, and (z) is the measure of another, this identical 
form of an (nth) power will express both parabolas, and 
the sum will be (y n+1 +z n+l )=2 (this conditional series), 
which cannot be a perfect (nth) power because of the 
factor (2). 

The proof lies in the fact that the identical form of 
two (nth) powers, and this form forbids the possibility of 
an exact (nth) root. 

In the general function of the (nth) degree, x n +Ax n ~ 1 -\- 
Bx n ---\-Cx n ~ :i .... -\-K, we have shown that, when the 
function is the product of (n) binomial factors, then the 
variables A, B, C . . . K, are successive derived functions 
of (K) , and, also, when, in the binomials, (x±a), (x±b), 
etc., (a) or (6), etc., is made a limit of (x) , then the 
function is = zero, for each limit. 

We wish now to show that (x n ), the basal element in 
(Fx), is identical with (K) . (x) represents* each limit 
of (K) separately and has one value. (K) contains (n) 
factors, each may be a limit of (x). In the general form- 
ula, series (1), (p. 8), (x) is the general limit of (Fx). 
(y) is the differential of (x) and — (x) . In (K) the 
factors a, b, c, are their own differentials, and (x) ap- 
proaches the limit (a) negatively, as (y) approaches (x) 
in series (1), (p. 8). (K) is a group of factors of (K) , 
or, limits. If we differentiate (K) successively, and inte- 

—41— 



grate (x) , as (y) is integrated in series (1), (p. 8), and 
combine by the same law, the result will be (Fx) in series 
(1), (P. 8). 



d K= 

d 2 K= 


K =K 

J x = Jx 
21 x 2 = Ix 2 


d z K= 


2 
2.SH x 3 =Hx z 


d*K= 


2.3 
2.3.4G x 4 =Gx* 


d 5 K= 


2.3.4 
...5F a- 5 =Fx'° 


d«K= 


...5 

...6E x 6 =Ex* 


d\K= 


...6 
...ID x~ =Dx J 


d*K= 


...7 
...8C X s =Cx* 


d»K= 


...8 
...9B x 9 =Bx" 


d i0 K= 


...9 
...10A %*« =Ax 10 


d"K= 


...10 
...11 a;* 1 = a* 1 ^ 




...11 



In the above {Fx) is the product of eleven binomial fac- 
tors. It may be so by simple hypothesis, but by the dif- 
ferential hypothesis (nx*- 1 ), the above is equal to zero for 
every limit of (x) contained in (K). 

For series (1), (p. 8), contains all the limits of (Fx). 
It therefore contains all the limits in (K) . Series (1), 
(p. 8) = zero on the hypothesis of an infinitesimal unit. 
(nx 11 * 1 ) is the hypothesis of the infinitesimal unit. It is 
true of all units. 

If A, B, C, etc., are not successive derived functions of 
(K) by the differential law, then (Fx) cannot be equal to 
zero for any ascribed value of (x) . 

If (e) be a limit taken and this limit do nc?t satisfy the 
above law, it is not a limit. 

Numeral values may be given to A, B, C, etc., that do 
not satisfy the law of successive derived functions of (K ) . 

—42— 



ThQ co-ordinates may be so located that the line does 
> not cross the axis of abscissas* 




Y 

AO=x. BO=x. CO=x. AO=— a. £0=+6. 

CO=-\-c. x=—a. x=-\-b. x=-\-c. x-\-a~0. 
x — 6=0. x — c=0. 
(x-\-a) (x — b) (x — c)=x :i ±:Ax 2 ±Bx-\-C=0. 

a\ |- — ab\ _.;:•• 

±A= | — b\. B=\ — as|. C=-abe, or a( — b) ( — c). 
| — c\ \+'bc\ 

See pp. 45-46 for fuller study of this subject. 

In the above general equation of the third degree, (a), 
(b) and (c) are variables, because they are the three 
limits of (x) in the general equation of the third degree. 
It is clear, that we may, b y moving the axis of abscissas 
up to (F) or down to (G), include all points of the curve 
lying between these parallels. (C), or its =OE will be 
variable, and (a), (b) and (c) will touch all limits of the 
line lying between the perpendiculars drawn through (F) 
and (G) to (yy). 

|+ a |- I — a ^| 

When A=\- — b\ or B,= \ — clc\, are numerals, or constant, 

[■ — c \ l+^ c l 

then the numeral value of C=(a) (— b) ( — c) is fixed or 

constant, ;* for when we substitute (—a) for (x) in the 
above equation of the third degree, we have x 3 -\-\ — b\x 2 -\- 

\-ob\ " ~° . 

| — ac\x= — a 3 -f a 3 — a 2 b — a 2 c-\-a°-b-\-a*c—abc. C=-\-abc, and 

limit (a) makes the function zero. 

The numeral values of the second and third coefficients 
of the equation determine the line. The third coefficient 
is the determinant of the distance from the origin to the 

—43— 



point on the axis of ordinates at which the line crosses. 
L is variable,, (A) and (B) constants. 

These two coefficients (A) and (B) , relate all points 
of the line to the coordinates, and thus the line is fixed 
and measured. In the above, (a), (6), and (c) are vari- 
ables of (x) . The differential of (a) or (6) is unity, 
(a), (b) and (c) relate the points they denote to (0), 
the origin. 

I fwe put BC, in the figure above, = (y), then x=AC 
and (x — y)=AB and two limits are defined so that when 
(x — y)=zero, x=OC. This has been shown on a preced- 
ing page. 

By referring to the above figure, which illustrates the 
function of the third degree, it will be seen that there are 
three values of (y). 

For any value of {x) , there are two values of (y) by 
which (x) is found, as shown (p. 10). In the function 
of the second degree there is but one value of (y) for both 
values of (x), so that (y) in the second degree, as a 
distance between limits of (x) , is constant, and, variable 
in the higher degrees. Now the only way (y), as an in- 
dependent variable, can be joined with (x) to form a 
function is as a determinant of distances between values 
of (x) . As a variable, (y) can be joined with (x) to 
represent all values of (x). (x — y) may stand for all 
the values of variable (x). 

x 2 -\-y 2 =z 2 is a line of the second degree as shown above, 
and only one of the two symbols in the first member is 
variable, the function y?=0, and with variable z=-ab de- 
fines all points in the line. (z 3 -f-i/ 3 ) is of the third degree 
and both symbols are variables. Now, if we apply this 
to Fermat's Theorem we have as follows : Hypothesis 
x n -* r y n =z n . Then 7ix n ~ 1J r ny n - '=nz n - . The increments are 
taken infinitesimal. 

The differential coefficients are ratios, or multipliers, 
and the increments approach equality or unity, and the 
members are equal. Divide by (n) and ^ n_1 -f v D - =z n ~ '. 
This proves that the sum of two (nth) powers is always 
an (nth) power. But, there are values of (x) and (y) in 
which this is not true; hence, the hypothesis is absurd 
and untrue. There are forms of x 2 -\-y 2 =z 2 that are true; 
but (x 2 -\- y 2 ) is not included, as has already been proven in 
the above hypothesis, as t (y 2 ) is constant, and we should 
have as differential coefficients 2x=2z. 

—44— 



v If we add the increments to the above equation; nx n ~ 1 A 
x-^-ny 11 ' 1 Ay=nz n ~ 1 Ax+nz 11 - 1 Ay. These are expres- 
sions of magnitude, not ratios, as the increments are con- 
crete, ifinitesimal, equal units. Now (x) and (y) are « 
at zero; and, so z=2 l and we have x n ~ ] J r y u -^^=2z n ~\ 

But 2z n ~ x is not a perfect nth power. So (ic n_t -|-2/ n — ' ) 
is not. If the form is not true! at the limit zero it is not 
true of any limit. 

This is the same result as reached in Studies (pp. 23-25). 

There is yet another method of proof, as follows : 

We have shown, (pp. 28"29), that the first differential 
coefficient is the number of combinations of (n) things 
taken (n — 1) together. This is when the variable is the 
unit of measure. Thus (nx*- 1 ) is [n(n — 1)] different 
combinations of one factor with the rest. Let a, b, c, d, 
etc., be the factors. If we measure by (a) each of the 
(n — 1) remaining factors will give a varying effect and 
there will be (n — 1) changes, and so for each of the (n) 
factors. Then the n(n — 1) is the total ratio of change. 
And this (n) must be multiplied by the product of the 
factors and we have nx n ~ 1 -\-ny n ^-=nz ll - } , if x n -\-y n =z n , 
which is not true. The combinations when the variables 
are made units of measure approaching equality at zero 
would be for each n(n — 1). The number of combinations 
in the first number would be just twice those of the 
second. So the symbol of value could not be (z), but 
must be 2*2?. No two independent variables can therefore 
be equal to a third variable dependent on the two vari- 
ables. 

Variables below the third degree in the same function 
are dependent. 

In the above discussion, when the function is assumed 
at the limit, or equal to zero, (x) must have zero. as its 
limit both in the function and in the differential coefficient. 
The equality is then true in the differential equation 
derived from the function x n -\-y n =z n . x n -\-y n — z n =0. 
nx n ~ 1 -{-ny n ~ ' — nz n - ' -=0. 

Either of these symbols x, y, z, can be = zero, if the 
equation is true. 

On (pp. 17-21, also on (pp. 28-30) and (pp. 43-45), 
the subject of the differential relation of the coefficients 
of the terms in the function of the (nth) degree is dis- 

—45— 



cussed. These relations are shown to be the law of 
series (1), (p. 8). It has been proven and illustrated 
in th£ above pages of study that (x — y) is a general limit 
of (x) in all functions of (x) and (x — y) n is an expression 
of all the limits of (x) . Also (Fy) includes the last term 
(L) in (Fx) =x n +Ax*- 1 -\-Bx n - 2 . . . L. The law of relation 
of these coefficients is the same as the successive derived 

coefficients of functions of (y n ) . 

• 

We will now in a general way show that this is true of 
the (FL) in the general (Fx)=-=X n -\-Ax n - 1 +Bx 11 - 2 . . . L. 
x nJ r Ax n - 1 -\-Bx n -' 2 . . . L=genera! (Fx) . This hypothesizes 
no limit for (x) and so the coefficients are independent 
variables and the {Fx) is not = zero although it includes 
this limit, and all limits as a variable function. 

Now put the coefficients constant on this hypothesis, 
and differentiate. 



xn+Ax^+Bx*- 2 .... +L 
nx n ^-\-(n — l)Ax u - 2 . . . -\-K 
+2/ 

+2.3/ 

....;....;. +2.3.4# 

-f 2.3.4.5F 

+2.3.4.5.6£ 

: +2.3.4.5.6.7Z) 

+2.3.4.5.6.7.8C 

+^.3.4.5.6.7.8.95 

+2.3.4.5.6.7.8.9.10A 

! . . . 2.3.4.5.6.7.8.9.10.11- 



(y) 

2 

ir_ 

2.3 
2.3.4 



We have omitted a part of this work in the terms con- 
taining (x). It may be supplied mentally. These are suc- 
cessive derived functions of (Fx) and sustain this relation 
for all limits of (x) . Put .>:=(), and these coefficients are 
the successive derived functions at , that limit of (x) . 
Since (L), (K) , etc., are independent of (x) in the general 
function of (x) above; when (Fx) is not = z$ro, then 
the above would be true to variable (L) at each limit of 
(L). 

In the above hypothesis that general (Fx) has the two 
independent variables, (a;) and (L), (x) has all the limits 
of (L), and (L) has all the limits of (x) . For each has 



-46- 



all limits. The whole must contain any hypothesis on the 
preceding page, if (x) = (x — y) , as on (pp. 6-7), repre- 
senting all limits of (x) , (when y=x) , then, by orftitting 
the factorials, which cancel, the original function is re- 
stored, and the coefficients are successive derived func- 
tions when the factorials are restored, derived functions 
of (L) . This proves in a general way that our method of 
differentiating (abed etc.)=L is correct because the 
result is true. When a limit is taken for (x), (Fx) must 
be=zero for that limit, (pp. 43-45). Series (1), (p. 8), 
measures all magnitudes with (Fx) at zero. The (Fx) 
is restored by the hypothesis (pp. 6 and 7). (£)=its 
differential. This is different from the usual hypothesis- 

The general differentials of all magnitudes are taken 
at zero, so that (F#)=zero. The above hypothesis gives 
this function for all limits and so is natural, and makes 
series (1), (p. 8jr, the- general integral law, and gives the 
natural analysis of (Fx). 

If A, B, C . . . L, are arbitrarily assumed so as not to 
have the above differential relation (pp. 45-46), then 
x n -\-Ax n - 1 -\-Bx n - 2 . . . L may be = to (Fx) for one or more 
limits of (x), or for no limit of (x) . It is plain that the 
line indicated by (Fx) would not be traced for any point 
when (Fx)=zero. One point or two or more may be on 
the axis of abscissas. 

(Fx)=zero for all limits of (x) will give (n) limits 
for the function of the nth degree when the above law of 
derived functions holds for all limits. 

(Fx) may not be an equation. 

x nJ rAx u - 1J \-Bx n -- . . . +L=Fx. As in the variables (x n ) 
and (y n ), (L) contains, as factors, all the limits of (x). 
In the formation of the function above the same law that 
develops (x-\- 2/) n =# n , develops the general function. The 
development of each term of the above function is the 
same as that of the series of terms. If we make the 
limits of (L) each = (y) , (Fx) will be the same as 
(#+?/) n . (y) is a general term for each factor in (L) ; 
and y n =(L). (A) is (ny) , (B)=n(n — l)i/ 2 /2, and so on. 
Therefore every function of (x) is differentially F(x±y). 
A, B . . . L are limits of (x) and also of (y) . In the 
above function (n) limits related to each other are in- 
cluded in (L). 

If (A), (B) . . . (L) are assumed constant, then there 

—47— 



are (n) constant limits. If these coefficients are variable 
so that they follow the law of successive derived functions 
then they are each measured by (x) and have the same 
differential, (dx). [x) has incommensurable limits. The 
general symbol of such a limit would be (a — \ b) . 

(x — y) approaches all limits of (x) has been fully 
shown and illustrated, (x) approaches the limit (a — \ b) . 
Now we can substitute (a — \ b) for (x), and make (a) 
and ( \ b) two independent variables with the same relatio i 
of (x) and (y) . —\b distance between limits of (a), 
or, (a) distance between limits of ( = \ b) . 




Let (bed) be a line measured by rectangular coordin- 
ates; (c) and id) two limits of 0:)=a = \ b, when (a) 
=oe) . ( — \ b)=ed. ( — \ b) = (ec) . When this ex- 
pression represents all the limits of (x), then (x) dis- 
appears, and we use the above figure to locate limits of 
(a) or { — \ b) . (ar=\ b) approaches (a); or ( = \ b — a) 
approaches (\ b). The penpendicular line at (e) is the 
axis of ordinates for ( = \ b) . (a)—oe, the distance be- 
tween the co-ordinates for (a) and ( = \ b) . But the in- 
finitesimal unit of the differential brings these two points 
together at (o) or (e) . At (o) when ( — \b) is differen- 
tial, and at (e) when (a) is the differential. All limits 
of (±\ b) are measured from (e) all limits of (a), from 
(o). (e) is a variable point, (o) is fixed. Thus (-- \ b 
— a) = (ed). ( — \b—a) = ( — ec). The differential (a) 
= ( — \ 6) at zero and (— \'b) is restored = (cd). 

Substituting (a — \ b) for O) in (Fx) we have: 

(a— a 6) n — A (a — \ 5) 11 - 1 .... -\-L. 

Let us use the fourth degree and eliminate the incom- 
mensurable \ b. 

(a-\ 6) 4 -A(a-\ 6) ? -5(a-\ b) 2 +C(a-f a b)-D. 

[4(a\ ^6) s -3A(a- > b)H-2ff(a+> &)-CJ(-\ &). 

— 48— 



[3.4 (a+\/6) 2 +2.3A (a+ VW +2#] (+6/2) . 

[2.3.4(a+y6)+2.3A](+y& 3 /2.3) (2.3.4) '. (+672.3.4). 

The above is series (1), p. (8), and is at infinitesimal 
zero. (a) is the differential. If we put + y6=0 and 
a=\/b, as above, and, also, that the (Fx) above has no 
constant element; and, when x=0, the function = 0, then 
what remains of the series will be zero, and we have 
(4cr+3Aa-+2£a+C) y 6+ (6a 2 +3Aa 
+B) y6 2 +(4a+A) 

If we divide this by (\/b), we have terms containing 
(V6) which may be left out as the remaining terms =0. 
4a 3 +3Aa 2 +2£a+C+4(a+A)6=0. 
&= (_4a 3 — 3Aa-— 2Ba— C) -f- (4a+A ) . 
__4 a 3_3A a 2 —2Ba — C_|4a+A. 
— 4a 3 — A a- — a 2 

—ZAa 2 —2Bar—C 

If A, B, C, etc., are intigers, then there can be no 
fractions in the roots and (a) and (b) are intigers. 

The criticism arises, why (Fx) above was not differen- 
tiated for variable coefficients. This would have given a 
result identical with that obtained, and if two identicals 
=0, then one of them=0. 

The above result is the first derived function divided 
by the third derived function when y6=0. 

In the above (Fx) , (Fx) is not zero because of the 
absolute hypothesis but when (Fx) is diiferential it has 
one of its variable limits in (L), and for any limit the 
function is equal to zero, (x — y) n represents (x) at all 
its limits. The expression is therefore equal to zero. In 
the infinitesimal hypothesis on (p. 7), it will be seen 
that, on this infinitesimal hypothesis alone, the differential 
law is derived, (wx n3 )- Therefore all functions by this 
law are equal to infinitesimal zero, as we started out 
en p. (5). The absolute zero and infinitesimal zero are 
represented in series (1), (p. 8). 

From this page above we take up the result: 
4a 3 +3Aa 2 +2£a+C+4a6+A6=0, for all values of (a) 
and (b) in the equation of the fourth degree. It is sup- 
posed that there are four incommensurable roots. It has 
been shown on a preceding page that (A), (B) , and (C) 
fix (D) . So it does not appear in the formula for values 
or limits of (x) . 

—49— 



Put A=4, £=—165, C=— 962. These three coefficients 

fix the roots of the specific equation. Then, 
4a : 4-12a- — 330a — 962— 4:ab+4b=0. 

There are two integer = also two integers=6. 

— ia *_12 . ■ - - ■ L <}a-h962 4a- i 

4 fl s — 4q-' — a- — 2c. --^-7. 1-326 

— Ba*+ 330c— 962 4.:- ~ =b. 

— 8a : — Sa 

338a-f962 

336a ,-(-336 

2a+626 

Whatever two values make the above remainder an 

intiger gives a value of (o), when (4u^4) gives the same 
number for (-\-a) and ( — a 1 ). The numbers are +7 and 
—9. 4x7+4=32. 4-.-— 9)— 4= — S2. These numbers, 
— 7 and — 9, substituted in the above quotient will give 
values of (6). 

There must always be two integral values of (a), in the 
equation, that reduces it to zero, when the corresponding 
values of (o) are substituted- There can only be two 
such intigers for (a) and two for (6) ; else, the equation 
would have more than four roots. 

Lei — . — ■: . — ~~ . — 1 •:■':. — . ': r ::~.r e:ua:: :/.. 
Then — 4cr — 18a-+114<i — 166 j 4a—-: 

— la 3 — 6a- — *&- — 3a4-132a — 166 



• "__.- — 114a 4uH-6 

-12a 1 — 18a 



—132a— 166 

The difference between 4—6 and — 4«Hl-6 is 12, 
4 (_|_2) — 4 (—5)= — 12. a=+2 or — 5, or we may have 
— 2 or -f 5. But since (6) is — , the negative (a) must 

be greater than -§-(«) in the root. We will then have 
— 2 — 2— f — f=.4=— 6. Otherwise, A= — 6. 

Substituting these values of (a), we find 6= — 3. ± 

=±\ — 3. .?=*-2=- — 3. ,= — •:=-. " 

We see that the imaginary root is included in the law 
of limits: yet no doubt here reached because of the real 
limit, (a-n\'b). This may not be true when all limits are 

imaginary 



Let A=34, 5=423, C=2362. Then 
4a 3 — 102a 2 — 846a— 2362 ^4a+34 
4a 3 — 34a 2 
" _ 68a 2 — 846a 
— 68a 2 — 578a 

—268a— 2362 



— a 2 — 17aH 2 68a— 2362 

4a+34 



The sum of the two values of (a) is (17) and both + 
in the coefficient. +a+a a =17. The values of (a) that 
render the remainder an intiger are ( — 5) and ( — 12) in 
the roots which is the sign of (a) in the formula. 
— 268a=+1340. 1340— 2362=— 1022. —1022 =— 73. 

14 
_a 2 — 17a— 73=— 25+85— 73=— 13=6. 
x= — a± V — 13. 

Substituting — 12=a, — 268a=+3216. 3216—2362=854. 
854^- (—48+34) =—61. —a 2 — 17a— 61=— 144+204— 61 
=—1=6. x=— 12± V— 1. 

The formula holds good for imaginary roots. If we 
were to put a?=a±V — b in the process by which the for- 
mula is obtained and divide by \/ — 1, — b would be elim- 
inated. 




The ordinate of the above line=zero for the four limits, 
D, E, F and G. When (OH) or the fifth term is variable 
then the function contains all the limits of the line. When 
D is constant then A, B and C are constants and only 
the four limits are included in (F.T)=ordinate. x= 
(x — y). If x=OG, then (x — y) may be -{-OF, — OE, or 
—OD. When ( X —y)=—OE t then (y)=EG. When 
(x — y) = — OD, then (y)=DG. These three values of (y) 
put (x) at the limit (G). Let (F) be the limit of (x) . 
Then (x-\-y) =limit (G). We have shown that when we 
have (+■?/) then we have ( — nx n ~ l ) and the differential is 
negative. When (y) = (FE), then (x— y) = (OF). But 
FE^(OF) — (OE). 

—51— 



We wish now to show that for each value of (a), or 
limit of (x), (y) has three values in the equation of the 
fourth degree or function of the fourth degree. On a pre- 
ceding page it has been shown that in the third degree 
function (y) has two values for each value of (a). Take 
the equation just solved with roots ( — 12 = a 1) and — 5 
= \ 13, instead of the imaginary roots. Imaginary roots 
are not real limits of (x) . In the formula (4a 3 -j-3Aa 2 -f 
2Ba+C— 4a&-hA&)=0, the imaginary element disappears 
with the radical —\ / b. 

The equation with real roots will be, a 4 4-34a 3 +395a : 
-f 1718a. .The roots — 12dtVl are real and rational, 
— 11), and ( — 13) ; the distance between them is ±2yi 
= (±2) = (y). For a=(— 12— a 1), we have —12 — 
A 1 subtracted from — 12+ a l=-j-2. 1=2= (y) For x= 
— 12+\ 1 we have(— 12— \'l) — (— 12-j-\ '1=— 2). 1=- 2 
=y. For a=(— 12— a 1), (— 5-a 13) — (— 12— \/l)=74- 
\ 13-M=(i/) or 7— V13+Vl=(y)- 
y^* +2 VI, or 7+V13+ VI, or 7— \13+V1. 
a*+34z 3 +395a 2 4-1718a+ (4a 3 -f 102a 2 -+-790a+1718) y. 
6a 2 +102a+39o (y 2 ) . (4a+34)?/ 3 . (l)y\ 

The values above substituted for (y), ( — 12 — VI), 
being substituted for (a), will render the sum of the de- 
rived functions=zero. Only the rational forms in the 
results are taken, as they are together=zero. The above 
is series (1), (p. 8). The function is=zero; also, the 
series. So the sum of the derived functions with (y)= 
zero. 

If we divide the above series of derived (Fx) by (y) 
we have 2/=8 = vl3. 
4a 3 +102a 2 4-790a+1718. 
(6a 2 +102a+395) (8 = a 13). 
(4a+34) (64±16V13+13) 

(1) (512±3.64Vl3+3-8xl3=\ 13 s . 

Now as the root is rational, ( — 11), only rational terms 
in y, y 1 , if are used, and the result will be the same for 
a 13) for both values of (y) . 
for both values of (y) . 

4a? s -H02a; 2 -|-790z-r-1718= - l02 

((itf+102x— 395) 8.83X8= 634 

(4a+34) 77 = —18x77= —1386 

(1) (512+312)= 824 

y=8+radical ?y 2 =77-fradical. 0000 

—52— 



>?/ 3 =824-j-radical. As the root substituted is ( — 13), 
multiplying by the radical part of (y) will give radical 
terms whose sum apart from the rational terms will= 
zero. They are therefore omitted and the sum of the 
rationals reduces to zero. The two limits to which ( — 13) 
is referred ( — 5±yl3) give one series of values for (y), 
ill') , (y*) > when the radicals are omitted. Dividing by 
the common factor (y) shortens the process. 

When £= — 11, we have y= — 2 or 6±yi3. 
4ic 3 +102.T 2 +790^+1718=46 =46. 
6.t 2 +102o;+395 =•- — 1. — 1><6 =—6. 

4a?"+34 ==—10. —10x49 =--=—490. 
(1) = 1. 1X450= 450. 

—000. 
When y= — 2 we have zero also. 

Let us use the incommensurable terms in (y), (y 2 ), 
(y 3 ) , etc. The function and derived functions with (y) 
are: 

£- t _|_34tf 3 +395; r 2 +1718£+. x =— 13, then 2/=8±yi3. 
4^ ;i +102x 2 +790^+1718 

6:t- 2 +102.r+395=+83. +83X 1=+ 83. 

It 4-34 =—18. — 18x16=— 288. 

1 = +205X l=+205. 



000- 
Again, t 4 +34t 3 +395t 2 +1718t+.t=— 11. y=6±\/ 13. 
4x 3 +102t 2 +790t+1718. 

6x 2 -f 102x+395 = _1 = _lx 1= 1. 

4^+34 =—10 = —10X12=120. 

1 =+1 =+121X 1=121. 



000. 

In the work above we use +yi3. If we change to 
— \/13> the multipliers become — , — , — , instead of — , 
+, +, and result zero as before. We might complete the 
illustration and application of the above given law that 
(y) has three different values for each value of (x). We 
could complete the work of illusrtation by making (x) = 
— 5+yl3 and {y) = {— 2yi3) or (— 6— yi3) or (— 8— 
yl3). Also (x)=— 5— yi3 and (y)+2 yi3 or — 8+ 
yi3 or — 6+y 13. For the four values of (x) in the func- 
tion given there are twelve values of (y) , different values, 
For ( — 2) and (+2) mark different points on the axis 
of abscissas. So also ( — 8+yi3) and ( — 8 — yi3, etc. 

The given function becomes zero when the distance is 

—53— 



~:—\-z. :-~'"~rz.\ :-. zzz::z.z. z:z.z zz. 
( — 11) and ( — 5 = \ 13) , also bdh 
'-■-: — :— 1: zz.z — : — 1: 
as, ( — 11) and ( — 13). In the (# 
are n(n — 1) values ;:' :z.z: 

zero- is z'zzz iizrrfur.i: :: 

::ii.r^ :.ss::k :z z'zzz 'Izzzizs :: 
:i : .'. t . .v. -z ~.z z.'.--'.'..- zzzzzzz m . 
.z. zz.- zz.:::: z. ' " z.-z. = 

of (x). See p. (45 

(x)„ or the first degree functi 

ii :::>,5zr z'z.z zz-:.- ::' i:sc:s53i 
ence. (x 1 ),. or the function of ti 

-r"Tr ~Z.- .ZZ.- ii IZISSIZZZ ZZZZ Z 

ii ::zznzzzz ::r ':::'-_ -z.Zz- 
:. .t ~z~" : zzzzmzzz z. zzz.zzz. z.:.r- : 
Let the limits of (x*) be a, 6, e 
«f„ €. /- Let ns suppose that («) ai 
Then we have (o) and (c) to fi 
(Fx) limited to (a) has values 
(Ff) limited to (<t?) has values 
(Fx) and (Ft) have no common 
:::::::::." ;.: z'zzz zzzz.r :h:.se~ N 
(e) and CO Kmit (s). Then ( 
(c) Bmit far) to (<f). (2) most 
:: :;.t :" : [izniTi : zz.z ' ~ji. 
: : r i — =: : s zz : : o: s.s : c I r z.z 3zz 



x), (wth) d< 

z-z.ZtZ :z.z 
The 
is a(-if- 



r_ii :z: f". 5:- 
: z. z if-rrf-f :r::r- 

Iz: -.-.t.t 

> . (y ) is variable. 
ie limits of (f 3 ) be 
) Bmit (x) to (<b). 
5 limit (<f). Then 
(*)„ (c) and (/). 

(6) and << 
ent- They are in- 
;:: — =: Lf: 

:. ":-.- . : .- _ T. ■ 
it of (x) and (ir). 



7i_-. :■:.-.. - 



v 



If (x 31 ) and isT) bs two variables so that (x) measures 
distances between limit? of Of), and Of) measures dis- 
tances between limits of (x), then no measuring" limit of 
eftbt-r fx) «ir (if) in either function will correspond to 
or be equ*! to any measuring- limit of (x) or (f ) in t\e 
other function, except when (x>=(#). 



2*-= Cx — /#> n cor (Lains 

Cf — — - 






z- z .zzzz ' -z :.f " — 1 
the limit of (x) in (x 33 ) not = 
then x=«, jF=&. Let 6, e„ if , 
of ($)„ each of which fixes ti 
etc. be the (it — 1) limits (x), 



lues of (#). 

■ >w pu: 
iff . Or) ; also, put Of} 
of If we take 

i bmit of Of) in Or), 
be the (» — 1) Kir 

a). Let I. m r n. 
: - " : - : : z.:z-- z zzz:\:~ 



- z ^ 



{k). Then (x n ) will be (a) with determining limits, 
b, c, d, etc. (y n ) will be (k) with determining limits 
I, m, n, etc. In these fixed limits of (x n ) and (y n ) will 
be all the (n) limits of (x) , and all (n) limits of (y) , 
but there is no measuring limit common. Let us make 
a common limit. Let (b) = (l). Then will (a) = (&), 
which is contrary to hypothesis, that is, (a) is not = (k). 
Therefore there can be no common limit. Let us suppose, 
that, when (b) = (l), then (c) and (m) remain un- 
changed and unequal. Then (c) and (m) would give, 
when substituted for (x) in (x n ) , and (y\ in (y n ) re- 
spectively (a) and (k) as before, (c) and (m) corres- 
pond with (6) and (I) and would be equal. So each of the 
(n) limits. Therefore (x) n and (y) n can have no com- 
mon measuring limit of either term when (x) is not = 
to (tj). 

When (%) = (y) then both terms have the same set of 

limits. 

Now put x n -\-y n =z n . Let (p) be a limit of (z n ) ; also 
b, c, d, etc., be the (n — 1) limits of (y) , measuring dis- 
tances between limits of (z) . Let (q) be a limit of 
(z n ) ; also (I), (m), (n) , etc., the (?i — 1) limits of (x) 
measuring distances between limits of (z) . 

Then (p) = (a), and (q) = (k). But (a) is not equal 
to (k). So (p) is not equal to (q) . And (z) has two 
constant values or represents two limits, which is impos- 
sible. 

Therefore, when (x) is not = to (y), (x n -\-y n ) is not 
equal to (z n ) . If (x) =(y), we should have (2z n ) . 

x n -{-y n =2z ri . 

This last has been shown before on (pp. 23-25), by the 
differential law, which gives two series of differential 
terms, identical, for each term in the first member of the 
above expression, showing that (;r n ) does not exist with- 
out (i/ n ), but they do not denote the same constant. The 
differential law takes (x) and (y) at infinitesimal zero, 
and so (x) = (y) as above and we have x n -\-y n =2z . 

One is a proof by variables. The other gives the proof 
when (x) and (y) are constant, having (n) constant 
limits. 

—55— 



LOCUS OF POINTS. 

A line is the locus of all its points. 




(a.v— b) is the law of the straight line referred to 
rectangular co-ordinates. The law of a line is the (Fx) 
which determines the position of all and each of the 
points of the line. 




a^-j-Aa^+Ba? 1 - 2 L=0. 

A, B, C, D, etc., except (L) are constant for a given law. 

(D) is the fixed point on the axis of ordinates. (L) = 
OD or D. (L) is variable. (Fx)=zero. XX changes 
position. 

DIFFERENTIAL PROOF OF FERMAT'S THEOREM. 

. .y r -=(y—x) n . y n is (Fy). (y+x) n is series (1), 
(p. 8), and both (Fy) and the series are equal to zero, 
when a limit of (y) is taken. Also .v r = (x-\-y ) n - In these 

— 56 — 



expressions, (x+y) 11 and (y-\-x) n , there are, in the first, 
when (x) is taken at a limit, (n — 1) limits of (y) . Also 
in the second, hwen one of the (n) values of (y) is taken 
as a limit, there are left (n — 1) of the (n) values of (x) . 
So there will be in both expressions, taken together, no 
repetition of any limit of either (x n ) or (y n ). We have 
shown before that (n — 1) limits of (x) n or (y n ) are 
never repeated for any other value of (x) or (y) . The 
differential expresses the ratios at each limit. The in- 
crements are equal at zero. But in this case we need 
not take the incremnt at or variabl at zero. For {y-\-x) 
= (x-\-y}. 

Now {y-\-x) n -{- (x+y) n differentiated on the hypothesis 
that (y) is distance between any two limits of (x) , and 
that (x) is a distance between any two limits of (y) 
gives n(y-\-x) n - 1 -\-n(x-\-y) n - 1 . 

These two expressions are now equal. Divide by (n) 
and we have (2/+#) n_1 + (y-\-x) n - 1 =2(x-]-y) Ti - 1 . 

Then (n — 1) may be any exponent. The second member 
is not a perfect nth power. (x-\-y) n represents one set 
of values and (y-\-x) n another set of limits which we have 
shown before is not = to the first one except when (x) 
= (y). If (%) = (y), there will be (2z n ) in the second 
term as before. 

Series 1, p. 8=zero. 

x 5 +Ax A +Bx ? '+Cx 2 +Dx+E. x=a± \/b. 

(5x 4 +4Ax s +3Bx-+2Cx+D) (y) 

10x*+GAx 2 +SBx+C) (y)- 

(10x 2 +4Ax+B) (y) s 

(5z+A) (y) + 

(1) (yY 

(a±Vb) 5 -\-A(a±Vb)*+B(a±:\/b) 3 +C(a±\/b) z +D(a± 
Vb)+E. 
[5(a±V^)*f4A(a±V&) 3 +3B(a;±v / &) 2 +2C(a±\/6)+D] 

(±V&). 

r+10 (a±y 6) 3 +6A(a±y 6) 2 +3B(a:±vM^C] b. 
[+10(a±V&) 2 +4A(a±y&)+B] (±V& 5 ) 

[5(a±VW+A] b 2 

(1) (±V& 3 ) 

In the above (±\/b) is the variable in series one. (a) 
is the differential (y). After canceling (+y6), the dif- 
ferential (a) is made =±V&. The function is developed, 

—57— 



and for the limits (a—\b) is =0. Taking the terms 
that are radicals, we have, after dividing by ( — \ o) : 

a 4 ^4Acr— 3Ba— 2Ca— D — . :■! ..: —4Ac— B > :— b-=0. 

When the coefficients A, B, C, etc., are integers, then* 
when =\ b is radical, (a) and (6) will be integers. T 
find the value of (6) in terms of (a) we have an equation 
of the secon ddegree, and (b) has two values for the four 
roots that may be incommensurable in the equation of the 
fifth degree. One root is an integer in the transformed 
equation with integers for coefficients. 

When we have (6) expressed in terms of (a), then that 
value of (a) that makes this expression an integer is 
equal to (o). 

b-+ (10a — 4A--^B) 6= — a 4 — 4Aa 3 — ZBa- — 2Ca — D. 

When A. B. C and D are numerals and constant, then 
the root will be rational for (6) is rational. (6) will b^ 
found in rational terms of (a) second degree. 




Let (DEFGHI) be anv line or svmbol of magnitude. 
Let (SS) be the axis of ordinates. Let (TT) and (TT 1 ) 
be two axes of abscissas, varying in position to include 
the entire line (DEFGHI). (O) and (O 1 ) are points of 
changing origin from which all points, or limits, of the 
line (DEFGAI) are measured. (OP) and (0*P) are 

rying distances from (P) to the origin, and are alw 
= the product of all the varying limits. Let (x)=ao, be. 

etc. Let (#)=a 1 o 1 . 6'o c etc Let {y m ) and < 
be the ordinates, respectively for these limits of (y) and 
. I. 



With (P), a fixed point, and (0) and (O 1 ) changing 
position along (SS), (OP) and (OP) variable, either 
equal or unequal, (x n ) and (y n ) will each include all the 
limits of (DEFGHI). The above has all been proven in 
the preceding pages, and also illustrated, and applied. 

Let (a l o\ b*o\ c\o\, etc.) be the differential limits on 
CPT 1 ), from Which (ao, bo, co, etc.) on (TT), as limits, 
are reached. a l o x — ao= — y — ( — x)=x — y=the limit (a). 
'CPT 1 ) is the variable line and approaches (TT) when 
the unit of (x) and (y) is infinitesimal. (y) = (x). x= 
(x — y)=0. x n =(x — 7/) n =0. This gives the limit (a). 
If we seek the limit (a 1 ), (TT) approaches (T^ 1 ), and 
we have (ao — a 1 o 1 )= — x — ( — y)=y — x> and (y) = (y — x) 
=0. y n =(y — o,') n =0. This gives the limit (a 1 ). (x-\-y) n 
=x n , (x-\-y) n =y n , at zero. The above reasoning applies to 
all the limits (b, b 1 , c, c\ cl, d x , etc.). From the above 
line, (DEFGHI), we have shown that the Fx=x n =0 meas- 
ures the entire line when Fy=y n =0> also measures the en- 
tire line. The hypotheses do not make (x) numerically = 
(y) . (x — y) is a limit on (TT), and (x) is a limit on 
(TT). (y) is the distance between these limits. 

(y — x) is a limit on (T 1 ^), and (y) is also a limit on 
(TT). (x) is the distance between these limits, (x) 
and (y) are independent variables when w>2, because 
there will be two values, or more than two, for each vari- 
able, as has been shown on preceding pages, and, also by 
the figure above, (TT) and the limits (a, b, c, etc.) vary 
in relation to the line (DEFGHI) while (a 1 , b 1 , c 1 , etc.) 
approach (TT). 

Also (TT) nad its limits approach the variable limits 
on (T'T 1 ). (T^ 1 ) is constant as to (TT) when the 
limits (a 1 , b 1 , c\ etc.) are approached; and then (x) is 
the differential, (y) is the differential when the limits 
on (TT) are reached. The Fx=(x — y) n = variable zero, 
gives two series of differentials. These series are identi- 
cal, and when n>2, if integrated, restore (x-\-y) n twice. 
This is as it must be, for we have shown that (TT) and 
(T 1 ^) represent independent limits. (2 n ) could represent 
only the limit (a). It could not have the value (a 1 ) at 
the same time, when (x) and (y) are independently vari- 
able. 

On (pp. 5 — ), and in all the previous discussion of 
limits, only (TT) is taken and (y) is distances from any 
limit to each limit. It can be seen that the figure on 

—59— 



page 58 and the discussion to page 60 agree with all 
that precedes. All the limits in CPT 1 ) approach (a) by 
the same law that (a 1 ) approaches the limit (a). So we 
have on (T^T 1 ) (n — 1) limits besides the limit (a 1 ) that 
fix the limit (a). So, also, each limit of CPT 1 ) approach- 
es the limit (b) on (TT). There will be (n— 1) limits 
besides (b 1 ) what become limit (b) at the same time. 
Each limit of (PT 1 ) has (n) differential limits in (TT), 
and the reverse. When the limit (a) is chosen in (DEF 
GHI), and (a 1 ) is chosen in (DEFGHI), then these lines 
(TT) and (TT 1 ) are fixed, and the (n—1) limits of 
(DEFGHI) are fixed. Any one of the (n — 1) limits 
will approach the (nth) limit, and so determines that 
limit as has been illustrated in the solution of equations on 
preceding pages. 

None of the (n — 1) limits in (T X T X ) can be equal to 
any of the (n — 1) limits of (TT). unless the (nth) limits 
are equal. 

If (x) is not = to (y), or (a) not = (a 1 ), then (x n ) 
never reaches any limit of (y n ) . If (z n ) is made to in- 
clude (x n ) , it could not include (y n ) . If (x) = (y), then 
x n -{-y n =2x ri . If (y) be constant, then its limit is (y) . 

x°--\-y 2 may =z 2 . 

For (a) there are (n — 1) varying limits. In all there 
are n(n — 1) combinations of limits agreeing with (nx n _ x ). 

x n -\-Ax n _ 1 -{-Bx n - 1 . . .-\-L=(x-\-y) n +A(z+y) n „ 1 -{-'B(x+y) ri -- 
. . . -f- (abcde, etc.) Each term=infinitesimal zero. 

In the figure (DEFGHI), (p. 58), we have introduced 
a more general illustration by taking a point or limit on 
(DEFGHI) outside of (TT). or not on it. This limit, 
(a 1 ), on (TT 1 ), may be any limit on (DEFGHI) as re- 
lated to anv other limit of the magnitude, and, of course, 
includes the entire line (DEFGHI). So that it includes all 
the limits of (TT). In this case (x) is one limit of 
(DEFGHI), and (y) is another limit of (DEFGHI). They 
fix the position of (DEFGHI) without regard to the axes. 
The law of (Fx) fixes the axes. With a changed position 
of axes the law changes the (Fx) 1 . With the general limit 
(a 1 ), Fig. p. 58, the distance between the limits is (x — ?/) 
for (y) is assumed to be the variable limit (a 1 ). But 
if (x — y) is the distance between limits, and (y) the 
variable limit approaching (x) , then the variable (y) and 
its differential (x — y) are always = and may be inter- 
changed. In Fig. (p. 58), (y) approaches (x) by the 

—60— 



> dif. (x — y) . In the former treatise of the subject, th<? 
limit (x — y) approaches (x) by the dif. (±y). The dis- 
tance between the variable limit and the fixed limit is 
the same. In Fig. (p. 58), two functions, independent, 
are considered — {Fx) and (Fy) . In one function the 
differential (x) is the distance (a) to (a 1 ) on (TT 1 ). 
The differential (y) is the distance from (a 1 ) to (a) on 

(TT). I r the figure (p. 58) one dif. is to the right of 
the limit and the other is to the left of the limit. They 
have contrary signs. This we have shown before in series 

(1), (p. 8),=0. 

We have not changed our notation in the discussion of 
Fig. (p. 58), for the reasons given above. The use of 
the notation shows it to be correct. The limit (a 1 ) in- 
cludes all the limits of any (Fx) or (TT). 

If n>2, then (x n ) and (y n ) are incommensurable, when, 
as functions at zero, they describe, each, (n) limits of 
(DEFGHI), as on (TT) and (TT 1 ),- (p. 58). Let 
(x) be the unit of measure at the limit (a). Let (y) 
be the independent unit of (y n ) . y=a x . (a 1 ) is not = 
(a) unless the limits b\ c 1 , d}. etc.=&. c, d, etc.; when 
(TT 1 ) coincides with (TT). and is the same line. The 
units (a 1 ) and (a) are never the same size when they 
measure (Fx) and (Fy) above the second degree. When 
7?<3, (y) is constant. There is no variable line (T 1 !^ 1 ). 
(x n -{-y n ) is not =z n . (z n ) canjiot be measured by an in- 
commensurable unit. z n =(a+\/b) n . 

Let (ABCD) be a square. 




(CB)=unit (a 1 ). (CA)=unit (a). Then let the lines 
approach zero at (A), (a 1 ) can never measure (a). 

When n>2, no two portions of the line of the (nth) 
degree are commensurable. 

Draw the third abscissa (T^T 11 )- (d\c l ) and (dc) are 

—61— 



incommensurable. (c^c 1 ) and {dc 111 ) are incommensur- 
able. 





* 


/c" 


Y 


r"r" 


df 


c' 




TT 


dl 


c 




TT 



The unit cf (ePc 1 ) cannot measure the unit df (dc 111 ). 
Let (&<?) = {$). (d 11 c) = {s); then \ (r-+s 2 )=dif. arc 
d^d\ Let (tfc- li:L ) = 00. (rf 1 c 111 ) = (f) ; then \ ' (u-+t-) = 
dif. arc (d^I). But (r) and (u) have units not commen- 
surable, v (r 2 +s-) and \0' 2 -K 2 )) are not commensur- 
able. (d^d 1 ) cannot measure {d l d) . 

Lines of the second degree are symmetric. Lines of the 
higher degeres are not symmetric. 

The differential of the variable measure (x) in the line 
of the second degree is (2x) . The differential of (2x) is 
constant. It is (2). The effect of the changing obscisso 
in the ordinate is (2.r). The ordi:iate=zero. 




[In these figures (a) and (a 1 ).. (&) and (6 1 ) represent 
position and distance.] 

Let (DEF) be a line of the second degree, (a 1 *) 7 ) and 
{ao) abscissae. The effect on the ordinate frcm (a) to 
{a 1 ), — (»)=a limit of U), (cr)=a limit of (;r), — is 
\2x). The change from (a) to (b') f or from (b) to (ft 1 ), 
is the same, (a 1 ) and (b l ) are similar points in the line 
(DEF). The change from (b) to (6 1 ) is identical with 
the change from (6) to (a L ). it is always (2x) . 

Now in the Figure (ABC) above- the differential is 
(3a; 2 ). The differential of this is (6x) . a variable. The 
law of change, (3x-) , has itself a law of change. No twn 

—62— 



Qrdinates can have a common measuring unit. Every 
point in the line of the (nth) degree, (w>2), is marked 
by an ordinate incommensurable wiht any other ordinate 
not on the same abscissa. 

(P. 58). Since (a 1 ) and (a) are not commensurable 
(OP) and (0 : P) are the products, respectively, of the 
limits in (L 1 ) and (L) of the (Fx) , and have no commen- 
surable unit, or limit, (a 1 ) is the general limit of ( 1 T 1 T). 
The difference between this theory of limits and the one 
in use lies wholly in this : that the unit of measure is the 
differential, at zero. When n>2, every limit of a magni- 
tude is measured or fixed by (Fx) at zero. (0) is not 
fixed origin, (x) and (y) are incommensurable zero. 

On (p. 28) is an analysis of the differential law. 
(±nx n _ l ). As that may not appear sufficiently clear in 
the notation and terms, we add the following: Let (a, b, 

c, d, etc.) = (L). (a, b, c, d, etc.) are limits of (x) on 
the abscissa at the point (0) in Fig. (p. 58). (a, b, e, cl, 
etc.), each represent a number of units. The unit in eacn 
is variable, and so, not commensurable. Let the collection 
of units in (a) be a unit of measure, concrete. Then 
(b, c, d, etc.) become discrete numbers and the product 
(L)=i(b c d, etc.) times the unit (a). Using (6), as 
(a), we have (a e d, etc.) times the unit (b) ; also, (a b 

d, etc.) times the unit (c). The sum of these products 
(b c d, etc.) (a c d, etc.) (a b d, etc.) (a b c, etc.) 
etc., is the magnitude (abed, etc.) though this gives no 
definite value. 

Now (a, b. c, d, etc.) are just units of magnitude. If 
each approaches the common limit zero, there will be 
then, at this limit a common infinitesimal unit, (1). The 
whole increment of magnitude of (a, b, c, d, etc.) will be 
(bed etc.) -f (aed etc.) +abd etc.) -^-(abc etc.) etc. But as 
the function = zero for any of its limits, and also the 
unit is infinitesimal and (a;)=zero. x=--(a) or (b) , etc. 
The variable measure (x) begins with its differential unit. 
Only one of the factors can be a unit of magnitude in 
the product, or a limit in any function at once; so the 
above sum of the four products is four times the function 
(abed, etc.), when one of the factors is the unit. If we 
make the factors, (a, b, c, d, etc.) equal, then dif. (x^) 
4x ? \ Int. 4x 3 =4^ 4 -^-4=a: 4 . We replace the number, or unit 
(x) , and divide by (4). Int. [(bcd)-\- (aed) -f (abd) -f 
(abc)] = (abcd) . 

—63— 



With bed) we restore (a). With (acd we restore 
(6), and so on- till we have 4(a6cd). Divide by (4) and 
this = ibed). The differential is concrete and refers 
numbers as measures of magnitude. The differentials of 

variable measures as m. < ;/> . ':'•. etc. are equal. It has 
been shown they are infinitesimal units. Series (1>. 
i p. Si. is derived from the infinitesimal hypothesis alone. 
[x±y) =:nfinitesimal zero. F I x±y I ^infinitesimal z *r : 
Dif. (af)=(l). Did. [y)= =('l) in series (1). (p. 8). 

When (;/* )=0 absolute, then [y) has all the values "' 
(jf). The function is restored, as shown in the integral 
law. Differentia] in this treatise is equivalent to the term 
zero-limit. It means the concrete limit zero as including 
all limits. The element of time or progress from one point 
:f magnitude to another is eliminated. 

If we divide (a) by an infinite number, the ratio-limit 

is zero. Then each limit is a measure of the function 

HHy 3 and the ratio-unit which is the same in all 

the limits, (a, o, c, d, etc) is the differential- and the unit ; 
and the reasoning on (pp. 62-63) will not be changed as 



to the resulting differential law. 
the results, which will be numerical 
(a 8) is an infinitesimal ratio. A 
limit. The unit is a ratio. Jr the 
tude. If we take the ratio [a B I 
includes the limit [a • as before. 
hypothesis (a S) to (a). 



— nx-~, nor m any ol 
and not concrete. But 

'1 ratios are = at this 
v it is a magni- 

as the unit, this limit 
We onlv remove the 




In the two general lines, (T'T 1 ) and IT), it has 
shown that none of the limits of {DEFGHI) are commen 



—6- 



surable when the lines are not identical, (y) is not com- 
mensurable with (x) as the differential of (x) , except 
when = (#)• All the limits of (DEFGHI) are marked by 
perpendiculars from points on (ao) . These points on (ao) 
mark positions of (ao) that have no common unit of 
measure with other points on (ao) , except, they may be at 
the limits of (DEFGHI) on (ao) . (ao) has therefore an 
infinite number of incommensurable portions. Any one 
of these will make the function (x n -\-Ax n ~ 1 -\-Bx li - 2 .../>) 
=zero with all the coefficients constant except (L). The 
Line (ao) may be so divided that every point will render 
(Fx)=0. Take the equation, (1) x'^+x 2 — 41# — 105=0. 
Use (+6) as a limit. (2) x 3 +x 2 — 41x— 6=0. (— L=— 6 
is a coincidence and is not of consequence). The roots of 
(2), are (+6), ± i/ 2 (3y5— 7). (—14/2+6)=— 1. Change 
the sign. A=l, as in (2). ( — 3, — 5, +7) are the roots 
of (1) above. This is no doubt the result of a law of the 
limits in equation (1) that relates its limits to those of 
Eq. (2). Put +*(3V5— 7)=— a, — y 2 (3y5— 7)=— b. 
+6=c. Then ab+ac+bc=p=—41. (a&c) = (6) ~L\ 
For this line of the (nth) degree, (A, B, C, D), etc. are 
constants and limits of the successive derived functions of 
(L) . a, b, c, etc., the limits, chnage numerically in the 
constants, while the limit is always the same. In the 
function (L) the limit varies. 

The successive derived functions of (L) are constant for 
a specific line. We have shown before that the derived 
functions of (L) are multipliers and not concrete, so that 
A, B, C, etc. are discrete numbers, and so (a, b, c, etc.). 
The measuring limit of (L) is concrete in its unit. This 
limit on the abscissa is concrete, and so, each limit, when 
it is the measure of the (Fx) or (FL) . 

The unit concrete in Eq. (1) has been shown to be in- 
commensurable with the unit in Eq. 2. The unit (6) is 
not commensurable with the unit (5) in Eq. (1). This 
is the basis of the proof of Fermta's Theorem. # 5 =6, 
will have five limits, or roots of (x) . y 5 =l will have five 
roots, or limits for (y) and none of these limits of (y) 
will have a common measure for any limit of (x) . 

x nJ rAx n ~ 1J rBx n - 2 . . . +L and x n -\-Ax n ~'+Bx n - 2 . . . L 1 . 

(L) and (L 1 ) are variable, each measure all the limits 
of (DEFGHI), p. 64. No limit in either function can 
ever be numerically the same as any limit in the other 
function. The concrete unit in the one can have no com- 

—65-^ 



mon measure with the unit of the other. B \ x varies its 
unit of measure with variable (?i) . All the successive 
derived functions of (:r n ) will take the form x ln a in 
which 1 — n is never divisible by (»), and so will have the 
same infinitesimal measure (x 1 Q ). 

If (y). distance between the limits of (x) and (?/), 
be measured by an infinitesimal unit of the (nth) degree, 
then will (x) have the same measuring unit at zero. Let 
-\ sc= n,i> \ y. Then n \'x n =^ h \y n . x= n \ y n . n ~ h \ i/ n is 
the (nznb)th root of (y n ). 

The expression remains the same infinitesimal of (Fy). 
(x) cannot approach the limit (y) , nor (y) , (x) . When 
( n \ y) is the measure of limits, then we have also tt \ x 
as the general limit. 

(.r n +A.r n - 1 ^B.r n -- . . . — L). Put ( n \ y) differential. ) 

[nx ul -\-(n — l)A.r n --— (/? — 2)B.r D — . . . ^K] B \«/. ) 
[,;(/<— lh- : -—(??— 1) (n — 2)A.r E - s -i-(w — 2) (« — 3)- ) 
Bx»- 4 . . . +/J n \ (y) 2 -^2 =0 

[»(n— 1) (n—2)x*- — . . . — 7] tt \ (u)*-^2.3 i 
) 

+v ) 

If this series of derived functions of (Fx) above, be 
completed by the law of series (1), (p. 8), and the In- 
finitesimals and commensurables separated, we hr.ve 
(x u -^Ax Til ^Bx Y] -- . . . +L-\- y=0). Referring to Fig. 
p. 64, the abscissa has changed, the distance between 
(O) and (0). (L) has varied to {L±y). The changed 
function above stands now for the limit (a 1 ), if (a) was 
the lmit before the change. The changed function has the 
differential value of (.r), for it has the value 
of (x) in the series above. As (y) in the second 
function is not commensurable with (L), so {L^y) cuts 
(L) incommensurably. Ti \y=(ak), (Fig., p. 64). (0 0) 
=the change in the ordinate at zero. The differential of 
(0*0), when ( r \ y) is made =x, is {nx nt ). This is the 
rate of change of the (Fx) at zero. (0 J 0) therefore 
agrees with the new limit (a 1 ) or (a) by the law of vari- 
ation. 

But we have not yet found the approximate value of the 
changed unit of measure in terms of that from which it 
has been taken by addition or subtraction. 

—66— 



DISCUSSION OF THE CONIC SECTIONS AS SECOND 
DEGREE FUNCTIONS, BY SERIES 1, P. 8. 




Lee (OPQ) be any circle, (0) the origin where the 
curve has a limit, (yy) the axis of ordinates, and (00) 
the axis of abscissas. Let (P) be the limit of the axis 
of ordinates. Draw (O^ 1 ) parallel to the axis of ab- 
scissas. ' 



-x 2 -f-2Rx=0, when the 
the limits of (x) , and 



The equation of the circle is - 
ordinate is zero. Let (x±yb) = 
\/6 be the differential of (x) . 

Then — (*+yfr) 2 +2R(;r+Vfr)=0 ) 

Series (1), (p. 8) [— 2(x— y&)+2R] y& )=--0. 

(—1) b ) 

—x-+ZRx—b=0. x-— 2R^+6=0. x=±\/ (R 2 — b) +R. 

Put \/b=AB or (OO 1 ). Then ±V(R 2 — 6)=CD. 
«+.y(R2_&)+R marks the limits (A). R— V (R 2 — b) — 
(VA 1 and marks the limits (A 1 ). (OP) = (L) in (Fx). 
— 6= (AD) or (OO 1 ), as explained on (p. 124). 

Let AQ=an arc of (30°). Then the sine AD=y&==(l/ 2 ). 
5=14, (R=l). CD=±y(l— %) = ±V%. Then limit 
(A) is 1+V%=0 1 A. 1— V 94— limit (A 1 ) = (0 1 A 1 ). CD 
=«'cosine (30°) =±V% =.8660+. ^=O x A = 1.8660+. or 
(0^0=1— .8660=.139— . The sine of (45°) is .70711-'- 
= v/6. ' Then ± V Tl— (70711+) 2 ] = (AB) or (CD)=±: 
yO_.5)=±y.5=(.70711 + )=cosine. (C^A) — (1.70711 
_|_). 1 A 1 =(1— 77011+)=.29289. So the equation — (x 1 ) 2 
-I-2RX 1 — b)=0 marks all the limits of the circle when the 
circle touches the origin (O). If we put the origin at 
(C) we shall have (— ^ 2 +R 2 )=0, and — x 2 +R 2 — 6=0 

—67— 



for all points of the circle. Without ( — b) the equation at 
zero gives two limits (0) and (Q). 

This is the same result we get in the discussion of 
(Fx) when (n)>(2), except that when (n) = (2), two 
limits on another abscissa are symmetric. See close of 
p. (62) where this subject is discussed. On p. (62), and 
following pages, series (1)- (p. 8) is adhered to in the 
symbols used and (x) stands for all limits and (y)=dir- 
ferential, and the reverse. 

(x) and (y) are not commensurable. }(x) and (\ b) 
have the same unit for four symmetric limits in all line-s 
of the second degree as will be shown in the discussion of 
these lines. 

Referring to (Fig.) (p. 67), let (OPQ) be sany ellipse, 

(yy) and (00) are the axes. The lines are similar co 

those of the circle. The equation of the ellipse is 

— (B- /A 2 )2Aa;— x 2 =0, when A=*CQ. Put (s+^ 6), a limit 

of (x) , and substitute. 

— (*+\ '&) 2 +2A(a+V&)=0 (A) = (CQ). 

[— 2(*+v&)+2A]v& i=0 

(—1) (b) | 

—x-+2Ax—b=0. Put yb==AD and it also = (DD\», 
th* limit of the point (D) in the circle. Hence — \ b will 
be the same in both circle and ellipse. x=±\ (A 2 — b)-f~A. 

If ± ^ (A 2 — 6=BD, then (BA)=±B. Ay (A 2 — b. 

#==Ad=(B A)V(A 2 — b). The abscissas (BD) and 
(BA) are proportionate to (A) and (B), (A) and (3) 
being the Vo major and minor axes of the ellipse. Two 
symmetric points are fixed (A) and (A 1 ). BD:BA::A:B 
— Geometry. (A)=one-half major axis. (B)=one-hali 
the minor axis. 

The equation of the hyperbola is, when the origin is ai 
the vertex of the transverse axis, and (F.r)=0, (B- A-) 
(a; 2 + 2 A^)=0. Put { x ) = (x+\ b) and we have (B 2 A 2 ) 
(x 2 +2Ax+b)=0. 

x=±y'(A 2 —b)—^. (A)=OC. 

In the circle and the ellipse (2A) is the limit of (x). 

A is limit of (\ b) . But in the hyperbola, the limit of 
(x) and (\b) is infinite, while (x) in (x 2 — 4A 2 =0) is 
constant. When (\ b) is greater than (A) it passes the 

—68— 



\ 




limit of (x) or (x) in the circle and ellipse, arid the radi- 
cal is imaginary. 

The curve has no limit on the axis of ordinates. (r) 
and yfr are not limited. Their difference is constant, (x) 
and (V&) have no limit. 

The differential of (x — \/b) is constant, (x) gains the 
same differential that (\/b) gains. Dif. (x — yb)=0. 
Series (1), (p. 8) requires ( — yfr)=zero. (x) and 
hyperbola on the hypothesis that (x — \/b) is constant. 



Series (1), (p. 8), requires (x- 
(V&) always have the same sign. 



yfr)=zero. ( x ) and 



The variable measure (x) and its differential (y) have 
contrary signs. The law of the hyperbola is (r — Vr 1 ) — 
(2A). (r—yr 1 ) 2 =4A 2 . In this, (r) and (\/r x ) both in- 
crease positively, at the same time; or, negatively, if the 
opposite hyperbola is measured. The development at the 
foot of (p. 68) is not true. 

(r) and (r 1 ) are the radius vectors of the hyperbola. It 
may be said that (x 1 ) a n d (\/b) are respectively portio n s 
of the axes, and (r) and (r 1 ) are radii. But the equation 
of the hyperbola referred to the center and axes is of 
the above form (x 2 ) = (D) 2 . If we separate (x) and have 
(x 1 — \/b)' 2 =D 2 f applied to the hyperbola, (a; 1 ) and (\/M 
are both positive or both negativ.e If one form is not 
variable neither can the other be. On the hypothesis of 
series (1), (p. 8), the function is not=zero, except at 
the transverse axis. In the ellipse, (r) and (\/r 1 ) 2 — 4A 2 

—69— 



=0, (r) and (Vr 1 ) each lie between (A)+ distance of 
the focus from the ce n ter and (A) — distance of the focus 
from the center, (r) diminishes as (V?* 1 ) increases. This 
is the differential law of (x) and (y) , in this treatise of 
limits. Series (1), (p. 8), develops the (Fx) at zero for 
each limit as shown in the ellipse and circle on pp. 
(67-68). 

In the parabola we have (r — y'r 1 )=zero. (r — y/r 1 ) 2 
=0. By series (1), (p. 8), this cannot be developed. If 
(V/' 1 ) be made incommensurable with (A), then we have 
r=V — ;*\ an imaginary term. The directrix is the center 
of the parabolic curve. And with this origin the ordinate 
is always equal to the radius vector, and when the origin 
is at the vertex (x 1 ) = (r — A); (A) being the distance 
from the directrix to the origin or axis of ordinates. 
(\/2i)x) is the ordinate, (x) has one limit- There is no 
differential (y)- (Fz) is not === zero for any value of (x) 
except x=0- In the conic sections the formulas are: — for 
the ellipse and circle (r+V** 1 ) — A 2 =0, for the parabola 
(r — Vr 1 ) 2 =0, for the hyperbola (r — \ / ?* 1 ) 2 — A 2 =0. A= 
diameter through the foci. In the parabola and hyperbola 
(?•) and (Vr 1 ) vary in the same direction to no limit' and 
are not differential. 

In the ellipse and circle (r) fixes two limits between 
which it varies. 

When (A 1 ), (B 1 ) are semi-conjugate diameters of the 
ellipse, then will x t =zt (B 1 A 1 ) [y (A 1 ) 2 — &]+A . The 
origin will be at the vertex of the conjugate diameter. 
The axes are parallel to the diameters, or, tangent to the 
curve of the vertex. 

In the above discussion x'b=AD (p. 67). \'b=* tht 



ordinate of (A) and 
formula y-=2Rx — z~. 



equals (y), the ordinate, in the 







/ ^-- 


"^S! 4 ^ 


/ r .-''' 


■ ' ' ' \\ 

C F'{ ] 


k •" 


'"""r-S^S'A 


\^___ 


5—-—^^^/ 


v_ 





p 
—70- 



% Put x=2A. x-{-\/b=2A limit Q. 

Put FA'=r and F 1 A 1 ==V^i- 

B 1 /A 1 )VC(A 1 ) ? — &L 
( r _|_y r= (2A) = (0Q)= limit (Q). 

( r __y r ) = (o) limit (D). 

(o;+V&)=(2A) limit (Q). 

(^_y&) = (0) limit (D). 

(x-\- V&)=Inf. limit in hyperbola. 

(. r _y&)=2A " 

( r+V r)=Inf. " 

(r— yr)=2A " 

(r+Vr)=Inf. parabola. 

( r _Vr)=0 

For the development of the study of the differential unit, 
see index. 

The differential unit is one, (1), as a magnitude, and has 
its limit at zero. All other limits of unity are arbitrary in 
the measure of magnitudes. We may make it any size. 
All units are equal as related to zero. Zero is a com- 
mensurable limit for them all. What is true of any mag- 
nitude, when its unit is at the limit zero, is true at any 
limit of the unit. Proportions are not changed. As a 
number the differential is a unit. The variable (x) , as 
a measure, represents any number, but as such is = (x/x) 

= 1. .4 

The unit, (1), has numeral limits when it is (x/x) = (l) . 
(x T1 =l) has (n) numerical limits. (x) = ( n \/l) . (x n — 1) 
=0- 

If we make ( n Vj/.) the differential of (x) , then no limit 
of (Fx) when (L)is constant* can be measured by ( n \/y), 
and the limits are all changed. The function is changed- 

(1) x : '+Ax n - 1 +Bx n - 2 +Cx n -' i +L| 

(nx n - l + (n — 1) Ax n ~ 2 -\- (n — 2)Bx n ~>+ (n — 3) 
Cx n ~* ...... +K n \/y 

[n(n—l).x»- 2 -\- (n—1) (»— 2)Ax n ~- -.+J] n V (2/ 2 /2) | 

( ,. 7)"V(?7 3 /2.3) 1—0. 



[n {n—1) (1)] n V(2/ n /2.3...w) | 

(2) {x n +Ax n -'+Bx n - 2 +Cx n -~ .... +L±i/)=zero. 

The limits of (x) have changed. (x) in (2) cannot 
equal (x) in (1). Subtract (2) from (1) and divide by 
(x — x)= xl \/y. Then, 

—71— 



a: n -rA.r-'-— Bj: t '-— C;r- S . . . — L=zero for any limit. 
.r n — A.r- : — Z?.r- : — 0°- n . . . — L— ?/=zero. Series (1). 
(a* — .r n ) —A (x v -' — ./'" ) —B (.r r -- — .r n - 2 ) — C(.r n c — a:"- 3 >- - 

y=o. 

Eac'i part is = zero when (a*) has its value. 

U 11 - 1 — .r- : .v:— . . . — o-^^-A (;c T -- z — x -:c . . . — .r z - 2 ) . 

. . . -fir— ^ r- 1 =0. 

If we put a:=a;, in (1 and 2), then "a y=zero J and i •■.-■'- 
_|_( % — l)Axi z -\- (n — 2)5.r n " 3 . . . 4-a. 

This is the first derived function of (1). and is the 
difference between (1) and (2). 

n A ?/=zero- Therefore eq- (2) is the general limit of 
eq. (1), and embraces eq. (1) and all the limits of function 
(1), the same as series (1). p. (8) has all the limits of a 
function. 

(y) is the differential of (1). and ~\ y is the differential 
of O). (x) t in function (1). is not commensurable with 
(x) in function (2). 

{x n -\-Ax v ^ L -\-Bx n - 2 -\-Cx nS . . . +L+y) is always = 
zero. It is the result of developing function (1). (p. 
71), by series (1), (p. 8). 

Series (1), (p. 8) is always infinitesimal zero. We 
have therefore proven, by taking the difference between 
(1) and (2), that that difference is the differential of 
(1). This difference becomes the differential of (1) when 
at zero. We have also proven the same by series (1), 
(p. 8). Therefore (2) is the general function of the 
(nth) degree, and is always zero- 

We desire now to apply this to the measurement of 
lines higher than the second degree- 




72— 



,(1) x^+Ax^+Bx^+Cx*-* . . . +L=0. (p. 71). 
(2) x n -\-Ax n -'+Bx u - 2 +Cx n -* . . . -fL+2/=0. (p. 71). 

Let a, b, c, d, e, f, etc., be limits on the line described by 
these letters, XX and YY axes. No. (2) includes every 
line and all limits of lines of the (nth) degree. It there- 
fore includes all the axes above when all the terms of the 
function are variable. If A, B, C, D, etc., to K f are con- 
stants and (L) is variable, and = to (L-\-y) , then the 
specific curve is described as above. On p. (27), it has 
been proven that the limits change with variable (L), but 
do not change A, B, C, etc. 

When (L) is constant, the function stands for the (n) 
limits, (b, e, h, m, etc.)- When we add (y) , and have 
(L-\-y) , then the line (b, e, h, etc.) becomes c, d, i, etc-, 
or, any line. 

If we use n V — V for the differential of (x) , then we 
have (2), p. 72) : 
(2) x n -\-A.x n - 1 +Bx n - 2 +Cx n - 3 . . . +L — y- 

When (n) is even, then n V — V n is imaginary. This func- 
tion, (2), has its unit infinitesimal because it is series 
(1), (p. 8). n V — y=x. x n =y, and both may be dropped, 
and Ax Ti -^-{-Bx n ~ 2 -{-Cx n -" . . . -\-L, represents the limits 
of (x) in (2). Now, if we put (x) equal ° _1 V — V> re_ 
peating the above process, Ax v — 1 -\-Bx n - 2 +Cx n ~ 3 . . . -\-L — 
Ay='0. When (n — 1) is odd, n-1 V — y is not imaginary, 
and (x) must be ( + ) to cancel ( — y) . The differetial 
and its variable always have contrary signs, (y) will be 
contrary to (x 1 — 1 ) , and we have Bx n - 2 +Cx n ~ z . . . L, which 
has the same value of (x) that (Ax n - l -\-Bx n ~ 2 -{-Cx n ' s . . . 
-j-L — Ay) has. If we reduce this till n — (n — 2) is reach- 
ed, then we have (Jx 2 -\-Kx-{-L=0- #=±V ( K 2 — L) — K. 

JU 2 7) Yj 

This gives the last value of (x)- Series (1) fixes only one 

Tnrt. This has been shown before in these studies. Now, 

referring to Fig., p. 72, as we reduced the degree of the 

equation by dropping out the first term we also dropped 

one limit from the line XX, and changed the position of 

the line (bO) to (cO^), and then to (jO 11 ) , or down to 

(gDn) , according as to whether ( — Ay) were ( + ) or 

( — ). If (L) is ( + ), then ( — y) is towards (O 1 ) and 

(O 11 )- When we have reached n — (n — 2), only the line 

(h, i, j, k, I, m) is included in the second degree equation 

—73— 



(Jx-+Kx-r L=0). Take (m) for the limit, which is 
changed, by dropping (x 4 ), to the point (e) . Then, by 
dropping AxPKto (k) . £ra= 4 V — y, tu= — y, nl= n - x \ — y, 
ku= — Ay. If (k) be the last limit, it is the value of (x) 
in the result [>= = \ ( K- — L) — K]. Now we may return 

[ (17 J) 2J] 

from this limit (k) to (m) , and so, back to the line (h, e s 
h, m) , and have one root of the original equation, (pp. 
27 to 30), also (pp. 64-65). The limits change as the 
axis of abscissas, XX, change position. The limits on the 
changed abscissa are not commensurable with the differen- 
tiated abscissa. L=the product of a changed set of fac- 
tors. 

But \x=± V ( K- — L ) — K] is a line of the second de- 
[ (4 J- J) 27] 

gree and has the limit (A:), in (1), common. The lines 
meet at (k)- (L) is the distance to (D) on the second 
degree line. (D) has changed position. (L) has the two 
factors of the second degree and is constant. One factor 
is also a factor of (1). When (D) is restored to its posi- 
tion on (DO), then (L) becomes by the change (DO), and 
the value (O ll k) changes to (Om) and we have a root of 
Eq. (1). We must now so change the value of (L) con- 
stant. We must add first (ku) . Since (ku) and (id) are 
infinitesimals they begin at (t) . (ku) begins both at (?0 
and (t). and (ku)=^ r 2Ay^ 1 \ l —y=x. (2A n - 1 Vi/) = (2) 
A [-r \\ ' (K 2 — L ) —Kl 2Ay=2A [+ V (K 2 —L )—K] . 

[-3(4^ J) 2/] [ (4/ 2 J) 2/] 

n_1 \ — y=(ul), the differential of (0 lx k) , and (k) changes 
to (Of and we have by increasing (L) by (ku) brought 
the point (k) to (0 in its Eq. of the second degree. And 
we have x=4)H. Our Fig., (p. 72) represents but two 
changes above (0), and one below. This would cover the 
solution of the equation of the fourth degree- There would 
be (n — 2) changes of the value of (x) to reduce the 
equation of the (nth) degree to the second degree- When 
reduced to the third degree, Ave would have Ix z -\-Jx 2 -\-Kx 
— L — ly. As above described, we drop to the second de- 
gree. a ,2 4-K+L=0. *r=+ \ (K- — L) — K. . Instead of 

- J F \ (4J 2 . J) 27 

(2Ay) as the increment or differential of (L) in the sec- 
ond degree- we have (n — 2)1, when (/) represents, not 
that specific order in the alphabet, but the fourth term 
of the function from the right. Supposing the Fig. (p. 

—74— 



72) to represent (n — 2) lines, XX, above (O), and the 
lines in Fig. (p. 72), were the lines that reached the 
second degree limit, then we should have (x) in the func- 
tion n — [n — 2). 

a- =+ y^T_L+7 [+VC£ 2 — L— JJQ"-^ ] —X. 
4P [ (4J 2 J 2J) ] 2/ 

2/ 
^+34z 3 +395a; 2 +1718a;+m6=0. 

In Fig. (p. 72), we pass, in differentiating the general 
function of the line in the figure, first, we pass from (h) 
negatively, as before, to (i) . (x 4 ) is ( + ) for the negative 
root. Then we go back to (g) for (x 3 ) is ( : — ) and (y) 
is ( + ). Instead of (2 Ay) we have (Ay) in the above 
formula. 

The roots of the above equation are, — 11, — 13, — 5 
±yi3. The root we get by the above formula approaches 
— 5+yi3, the least negative root. 

When (w=4), we have (x i -\-Ax v,J r Bx' 2 -\-Cx J r L. 



a; 4 +14a; 3 — 33a; 2 — 514a;+1332=0. The roots are — 2± 
y'41 f _5±V61. The least ( + ) root is (— 5+y61). Sub- 
tituting the numerals in the equation in the formula 
above, we get £=2.S212+. — 5+ y 61=2.8102+. Illus- 
trating the solution by Fig (p. 72), we differentiated 
negatively from (m) to (k) , and so get a positive root. 
4 y — y and ( — y) were used in reducing the degree of the 
equation. The limit (m) is found, for (DD) is fixed by 
(L), and the line (g, h, i, j, k, I, m) is fixed by the co- 
efficients, and the limits of the second degree must be some 
limit of the above equation. If the limit (p) were reached, 
then (— D— D) would be (L) . Only (DD) = (L). (L) 
and the coefficients fix the limit (D) and distinguish it 
frcm all other points in the line. 

Hence whc£ we differentiate negatively we get the least 
positive limit of the equation. If the equation has all 
negative roots, the differential will be positive and we will 
return to the negative limit negatively. (L) will be suc- 
cessively diminished to reach the limit (h) , Fig. (p. 72). 
To illustrate this, take the equation of the (4th) degree, 

—75— 



used on (p. 52 » . Irs r:ors are negative. 
On p. 72. we have function (1) and (2). — 

(1) x-—Ax--'-—Bx--- . . . — Ix — J -—Kx—L=0. 

(2) x n -\-Ax*-*-\-Bx n - s . . . -hIx*-\-Jx--\-Kx=L=-\ — l n =0. 

We may make (;r n -p\ — 1 -)=0. when the coefficient is 
(4-) and (n) is even, by adding minus twice the second 
term. The couplet will then be divisible by (ir-{- V — 1)> 
and ( n \ — 1) is a limit. When the coefficient is negative 
and (n) even we add twice the second term of the couplet 
and get a like result. For other signs and values of (n) 
the couplet is equal to zero. 

If we drop (.?;-— r \ — 1") out of (2) and obtain the dif- 
ferential function of Ax n - 1 -\-Bx n -- . . . -j-Jar-j-Jx — Kx 

—L. we have. 

1 3 > Ax-— Br-- . . . Ix—Jx-—Kx—L—A-\— 1^=0. 

Making A - '—A~-\ — l nl as before = zero, and omitting 
them, we find the third differential function (DBx-- . . . 

—I;r—Jj:-—K:;:—L—B^, —1'—. In rhe same way. by 
successive sreps. we obtain (fi — 3), / — J ..:-— K: ;; — L— I 
~~. — 1=0. These successive differentials are related to 
each orher as ill and i2». (2) has all the limits of (1) 
when E \ — 1*=$. (3) has the limits of (2) w i: 

.a — r- : =o. 

(4) has all the limits of (3) when B a \ — l a - 2 =0. (n— 3 | 
has the limits of (n — 4) when I m \ — 3 3 =0. Jx--{-Kx-\-L 
=0 has a limit of {n — 3), when Lr—I~\ — 1 3 =0. If these 
terms have the same sign and plus, bv adding 27 S V — l 3 
to Jx-^-Kx+L we have Iaf+Jx-+Kx+L±I n V ( — lK 
which is (n — 3). If (/) is minus we add — 2I n \ — l 3 . If 
~///' 3 -f-/ n y ( — l) 3 ] have contrary signs, then we add or 
subtract (I n \ — I s ) and have (n — 3). /x 3 -f-/x 2 +Jfcc+L-f 
I D Y — I s is equal to Jx--\-Kx-\-L when the differentials 
( n V — 1) and C-\ — l 3 ), have the same limit i 7 —I 

( n \ — l)=x is found in iJx--\rKx-\-L). This is sub- 
stituted in (w — 3), by raising this to the third power and 
multiplying by ±1) and adding to (L) gives ./.?:-— Kx- L 
z-l "\ — lh (or —2I'\ — 1 i. equal to Ix —7 : -Kx—L. 

In the same way we pass to Hx*-\-Ix*-\-Jx z -\-Kx-\-L. 

A. — A '■■-. — l n must be made equal to zero numerically 
to agree with (Ax— A'\ — 1=0). We then have in the 
rction both infinitesimal zero and absolute zero. 

The above, in formula, will be: 

—76— 



(1) x^-j-Ax^-j-Bx 11 - 2 . . . +Ix 2 -J r Jx 2 +Kx+L=Q for every 
limit of (L) constant. 

(2) x*+Ax n r x +Bx-* . . . +Ix*+J&.+Kx+L+*\/—l n ==Q. 
all limits. 

(3) Ax»~'~\-Bx»- 2 . . . +Lr 3 +J r a; 2 +Ka;+L==0. (L) 
constant. 

(4) A&^+Btit* . . .Ja5*>/a? 2 +Ka;-fL : fA n V— i"" 1 " °- 
All limits. 

(5) J5x ,n - . . . +J« 8 +/a; 2 +lfa+L=G. (L) con- 
stant. 

(6) Bx 11 - 2 . . . +Ix 3 +Jx 2 +Kx+L+B n \/— l n " 
=0. All limits. 

(w— -3) . . . Ix*+Jx 2 +Kx+L=0. (L) con. 

(ti—2) 7^ 3 +/o: 2 +K^+L+/ n V ( — l) 3 -= 

0. All limits. 
(n — 1) Jx 2 +-fiia;+L=0. (L) con. 

When the differential of the ordinate is added to No. 
(w — 1), it becomes No. (w — 3). When the differential of 
the ordinate is added to No. (n — 3) it becomes No. (n — 5). 
Likewise No. (7) becomes No. (5), No. (5) becomes No. 
(3), No. (3) becomes No. (1). 

We will now show by Fig. p. (72), the reasons why 
(Bx n ^ r B n \ / — l n ) may not be equal to zero. When n V — l n 
is even and (B) is (+), then -\-L passes below the axis 
of abscissas. No part of the line for (+L) lies below this 
limit, so to put the term -\-Bx u in line with the limit of 
the changed function we add minus (Bx n ) and that brings 
us to (L) minus (Bx n ) a second time puts the limit with 
(L—B n \/ — l n ), the limit of (x) on the axis of abscissas 
for changed (x). It is then equal to zero for the limit. 
If (Bx n ) is minus, then if (ri) be even, we have ( — Bx n — 
n V — l n )= — Bx n — B n \/ — l n . This is the same as 
(Bx n -{-B n \/-\-l n ) . When we have the negative root ( — x) , 
we have ( — x+ n \/l). When (n) is odd, we have — Bx a -j- 
B n \/-{-l n . When (n) is even, -5^ n +Z? n V+l n - When, 
( — B) and (ri) are odd, we have J r Bx n -\-B n \/ -j-l n . If (n) 
is even and ( — B) we have — Bx n -}-B n \/ J r l n . The 
couplet of equal terms with contrary signs (or the quad- 
ruplet), is a function of itself apart from the function of 
which it forms one separate part, and is thus independ- 
ently equal to zero for the same constant limit. When the 
differential term of the couplet, (Bx n -\-B n \/ — l n ), lies be- 
yond the limit of (L) it is imaginary and must be placed 
within the limit. For the third degree function there are 

—77— 



two costant values of (L). Each is the basis of limits be- 
tween two points on the axis of abscissas on the same side 
of the base with (L) . Limits on the opposite side of the 
axis are measured by ( — L 1 ). (L 1 ) may be found by put- 
ting the first derived function of (Fx) equal to zero. The 
roots will show the position (L) 1 . When (L 1 ) is found, 
it must be substituted for (L) in the function, the other 
coefficients remaining unchanged, and the nearest limit 
to (L 1 ) may then be found, and so all the limits of the 
function. This may be applied to the fourth degree func- 
tion. There will be a third degree derived function and 
three values for (L), )L), (L 1 , (L 11 ) . The roots of this 
derived function may be found by the formula. This pro- 
cess reaches each limit of the function of the (nth) de- 
gree. The sign of (L) as related to its differential 
[ n y ( — l) k ] will be discussed in the solution of problems. 

The imaginary expression means that we are beyond 
the limits of the independent differentials (x) and (y). 
(y) is then not at zero with (x) , nor (x) with (y) . 

In these studies (x) has been used as the symbol of in- 
dependent variable, or measure, of the {Fx). (y) has 
been generally used as the differential of (x). (x) being 
taken at absolute zero, the differential (y) is alway s= ( a- ) . 
The differential function is = the function itself. It is the 
variable function. Two symbols are used because change 
of magnitude implies change of limit, and two limits are 
considered, (x — y) is one limit of (x) taken at zero, both 
numerically, and infinitesimal in unit, and the limit (x) is 
the variable limit reached by the change of the unit from 
zero to any magnitude. This has been proven and illus- 
trated frequently in these studies, and is the foundation 
principle of serise (1), p. (8). 

When (a, b, c, etc.) are taken as specific variable 
measures of a function, as has been illustrated in previous 
pages, then (x) becomes the variable which = each of 
these limits, (y) is not needed in the series (1), p. (8), 
when (L) = (a.b.c.d, etc.) is developed, (x) is then the 
differential. More than two limits are fixed by this law. 
We now wish to show the meaning of the (unit), (1) in 
these studies. Then we will take up the relation of this 
unit to hte symbols [a, b, c, etc.) in (L), {A, B, C, etc.), 
as final term of (FL) and coefficients. (1), as a unit, is 
the symbol of magnitude, variable size of measure, as (1) 
yard (1) foot. Or it denotes size of number. (5) may 

—78— 



(5) the unit of measure, or (7), etc. (1) is a, b, c, etc. 
% It is the only variable symbol of number and magnitude. 
It has been proven to be the differential of both number 
and magnitude. The differential of (1) is the infinitesimal 
zero, as an expression of magnitude. This hypothesis 
alone gives (-tnx*- 1 ) as the differential law. The differ- 
ential of (a, b, c) , as variables, is (1) in each case. But 
this is a variable unit, or (1). (1) is not a number, ex- 
cept as it denotes a single unit, as constant in magnitude. 
It has none of the power of a number. If we attempt to 
give the unit (1) the power of a number in any formula 
expressing numeral relations, without fixing its relation 
to number first, the formula will not be true. To illus- 
trate by the problem [1/aj+l)], which has been discussed 
in these studies on a preceding page. If we divide, 
l^-(l-\-x) =*!-{- x — x 2 -\-x 3 -}-. This is not a true quotient. 
If we use (x) as the divisor of (1), then l-f-(x-j-l)'= 
1— - 1 -j-JL_+- This shows that unity cannot be used as a 

/y /y*'2 /y»o 

measure of (x). It is the differential of (x) . (x) can 
be /included in (1) as its limit. If (x) be taken greater 
than (1) the first series has no limit. If (x) be less than 
(1), the series below has no limit. If (x) =*(1), then the 
quotient is {Yi) • The (1) varies with (x) . It is plain 
that, if we divide the variable (x) by one of its numeral 
units the quotient is not limited when the remainder after 
division always contains the variable whose differential is 
confused with its unit of measure. The constant limit 
of (x) is made one of its units. Variable (1) is confused 
with the constant (1). Let (y) or (x) be changed to the 
variable symbol (1), we have (a — 1), (b — 1), (c— J ) 
approach, each, respectively, (a, b, c) . We have shown 
before that (da)=H°=l. (a n — 1)=0; then (a+ n V — 1) — 
zero. ( n V — 1) is the differential of (a). Divide and we 
have: 
a n — 1 (q-py — 1 

a n-l_ a n-2 ny( l)-fa n - n \/ ( 1 ) 2 — tt n " 4 n \/( — l) 3 

. . . ± n V(— I) 11 " 1 - 
[a n -f n y( — l) n is taken when (a) is ( + ) and (n) odd, 
or ( — a) and (n) even.] 

The above division is exact. But "V ( — 1 ) — (<*) 
at zero, (a) and ( n V — 1) have opposite signs. — (differ- 
ential formula ztnx^ 1 ) . In the above quotient the terms 
are all ( + ). The negative terms are ( — )( — ) = ( + ). 

—79— 



The (n) (nth) roots of (1) are the relative values of the 

(n) (nth) roots of (a) when (a) is, with its differential, 
equal to zero. When (a n — 1)=0. If we have (a n +l) 
then (a n ) is ( — ) by the differential law. If we take (a) 
= to one of its units, and (1) to be also a unit of (a), 
then we have (1 — 1), a constant expression, that has no 
relation to any number. (1) cannot be taken to be a 
unit of (a) . 

(a n — l")-r-(a — 1) gives (a^-f a r - 2 +a n - :3 . . . -fl). This 
quotiet has no limit of (a n — 1). If we have o^+Aa; 11 - 1 -!- 
Bx n - 2 . . . -\-L-{-y, and n \ y be n V — 1, then we have, 

( D V — 1 is always — VI). U'+ n V — l) n +A (x+ n \ — l) n -' 

. . . I/+ ( — l)=zero. Put (x) = (a), one of the in) 

limits, and (a+ n \/ — l) n -fA(a+ n \/ — I)"- 1 

+ {abcd, etc.) + ( — l)=zero. In this last, (x) is the dif- 
ferential =4 D V — 1. (x) approaches (a, b, c, etc). a n -f- 
n V ( — l) n =0. (a n -f n V — l n )^-(a+ n V — 1)= (a nl — a a - 2 
n V — l+a n - 3 n V — l 2 — a n - 4n \/( — I) 3 .... ±a n -t n '^ 
n \/( — l) n - 2 ±a n - n n \/( — D n " 1 =0. If 00 is even, then 
the last term of the uotient will be (4-) and the differen- 
tial exponent will be (n — 1) and odd and ( — ). When 

( n V — 1) is infinitesimal, then we have (cr 1 ) in the quo- 
tient. The quotient has {n — 1) roots of (a n ). These 
roots are not — to (a) when (a) varies, when (o n ) becomes 

(a n — 1). 

The same results will follow A[a n - y ^- n \ ( — l) Ii - 1 ]-^-(a4- 
n V — 1), or #[a n - 2 + n \/( — l) n - 2 ]-=-(a-f- n V — 1)» and so on 
for the entire function. For the root ( n V — l) = (a), each 
couplet will be equal to zero. [a-fA n V — l)]bcd, etc,= 
zero. To obtain A[a n - 1 +( n V — l) n_1 ] the function has 
been differentiated a second time. To reach B[a n ~ 2 -\- 

( n V — l) n ~], the original function has been differentiated 
a third time. To obtain K(a+ n x — 1), the original func- 
tion has been differentiated (n) times. If (r) be the num- 
ber of times the original variable has changed by adding 

( n V — 1), or subtracting n \ / — 1, from (a=o) , then the 
constant (a), or n \' — 1 restored, will make the term in 
the original function £[K n y( — l)"^]. 

We have not used this reduction with the second degree, 
for ( n V — 1) is consta n t in the second degree, as has been 
proven before in these studies. We have taken (Jx 2 -\-Kx 
-\-L) as a term of the series to which the function has 
been reduced by successive differentiation. Now when we 
reach a value of (a), or ( n V — 1), which renders the series 
zero, we have a root of (x) in the original equation, (a — x) 

—80— 



will divide the function, and (a) is a root. The series 
Jx*+Kx+L . . /( n V— I) 3 - • .C( B V— l) n - 3 +#( n V— D n - 2 
+A ( n V— l) n_1 + ( n V— I) 1 — Fx=0. Jx 2 +Kx+L=Jx- 

-\-Kx-\-L . . 7(cr-|- n y — l 3 ). ( n V — 1) constant is a root 
of both members of this equality. n V ( — l) 3 is a link be- 
tween (Jx 2 +Kx+L) and (Ix*+Jx 2 +Kx+L±I n \/— V). 
When we have added / n V ( — I) 3 to the second degree equa- 
tion, we have a root of (Ix 3 -\-Jx 2 -\-Kx+L) . This equation 
has a common link in the changed value of ( K V — 1) that 
passes to the next degree higher, u n til we reach (x n ) , or 
x n -\-Ax n ~ 1J \-Bx n - 2 . . . -j-L. We then have, in the second 
degree equation, with L -J- the added values as its final 
term, a value of (x), in this varying second degree equa- 
tion, which is the same as one value in the original (nth) 
degree equation. 

We have shown in connection with the general line how 
this must be the nearest limit to the ordinate (L). It will 
be the least ( + ) root, or the least ( — ) root numerically. 
The varying ( n V — 1) is negative when the term considered 
is positive. The variable ( n V — 1) is positive when the 
term considered is negative. 

The study from the (80 p.) may seem to be a repeti- 
tion of that begun on p. 76. The latter is wholly a 
differential proof by series (1), p. (8), that x n -\- n \/ — l n =0. 
A[x n - 1J r n \/ ( — l) nl ]=0, and so on, dropping by this 
equality, the terms of the function successively. It is 
also shown that ( n V — 1) is a common constant for two 
successive expressions, by which we may return to the 
function. The study from (80 p.) proves by division 
that all these terms, — (a n + n \/ — l n ), A[a n " 1 + n V( — l) n_1 ]. 
etc., are divisible by (a-\- n \/ — 1) ; and so have a common 
root. We then give (/£ 2 +K,x-f-L) a value for (L), by 
addition, which £ives one of these values of ( n \/-—l), or 
(a). 

The method by division corroborates that by series (1), 
(p. 8). The two studies als® contain other different mat- 
ter, and are only alike as one course of study corrobor- 
ates the other. 

If we differentiate, (L) = (a b c etc.), with (x) as 
the differential symbol, approaching these related but 
variable limits, as on (pp. 26-46). 

We have: 

—81— 



L=,L 


(1) 




L 


dL=K 


(x) 




Kx 




d"L=2J 


(x 2 /2) 




Jx- 




d 6 L=(2.3)I 


(z 3 /2.3) 




Ix 3 




tf 4 L= (2.3.4) H 


(x* 2.3.4) 








d»-'L=[1.2...(n- 


-2)] B x»* 


) Bx n - 


* 


)=0. 



(1.2.3... (w—2 ) 
d» l L=[1.2.3..(n— 2) (n— 1)]A (_ x*- 1 ) 

(1.2.3.. (n— 2) (?z— 1) 

d n L=[1.2..(n — l)(n — 2)?i]( .r n ) ..x n ] 

[1.2.3.. (w— 2) (n— l)?i] 

This is Series (1), (p. 8), and = zero for any limit, 
(a, b, c, d, etc.). Put (x), the differential of limits, in the 
Fia, b, c, etc.), equal to ( D \ / — 1), and the above result is: 

( n V— D n + A( n \/— l) n_1 + B n \ — l) n " 2 

+/( n V— D 3 4-/( n V— 1) 2 +X(V— l)+L=zero. (x») can- 
not be equal to ( n \ / — l) n when the latter is its differential; 
neither can (x) be equal to ( n y — 1). For (x) and (x™) 
are variable points of separation in value between the 
variable and the symbol of a changed value, and they are 
by hypothesis only equal when the unit is infinitesimal. 

When the unit, (1), has value, or separates from the 
zero limit and begins to give concrete value to (x)=0, 
then the valuers, ( n \/ — l) n , ( n \/ — l) n ~\ etc., are the ap- 
proximate values of (a* n ), (x n_1 ), etc. If a numerical value 
for ( n V — 1) be found rendering the same result as that 
when the symbols are at infinitesimal zero, then that will 
be a limit of (x) . ( n V— l) n , (V — l) nl . . . n \/ — 1, are 
incommensurables. ( n y — 1) is the symbol of concrete 
value or number. 

In the development of F(a, b, c, etc.), above, the 
series 1, x, (x*/2), (x* 2.3) . . . (x n 1.2.3. ..n) is an inte- 
gral series to the right, and a differential from the right 
to the left. (When written horizontally instead of perpen- 
dicularly). (.t b ) is restored in the resulting equation. 
When the series (x n , x n ~ l , etc.) is differential, then the 

terms ( n V— l) n , ( n V — 1)*"\ ( n V— l) n_2 

(*V — l) n -f n - 1 ], .are related as ffoQlows: (*V — l) n , 

n (n^_i)n-i f n [( n _i) f-y_i)]»-a f n [( nr _i) ( n —2) ( n V 

— l) n - 8 .... n(n— l)(n— 2) . . . n— (n— 1) (»y— 1). 

In this series the factorials of any term are the same 
as the term preceding it, except the next higher numeral, 

—82— 



which must be given its factorial power before adding 
to the next power higher. When added, this approaching 
value has the power of the product of the remaining fac- 
torials. At each step in approximating the true value 
of (x), one factorial drops out so that each term has the 
power of its own factorials. Thus this seriee on (p. 82), 
with factorials restored, reaches, by approximation, the 
value of (x) that satisfies the function. A positive root 
is found when the differencial is negative, which is the 
law of series (1), (p. 8). The terms of the function can 
have alternate signs changed if the roots are all negative. 

In these studies distances between lines and points in 
the figures are conceived as infinitesimal or not appreci- 
ables. They are illustrations of incipient magnitude. The 
formulae must be considered as at absolute and infinites- 
imal zero. When the variable =H absolute zero, then the 
infinitesimal takes all values. 

a*-|_6# 3 — 21# 2 — 62#+18=i0. Roots + (2±y3), — (5=n 
V7). 

A=+6, £=—21, C=— 62, £>=+18. Least + root + (2— 
V3) =.2679492+. Root found by formula, .26792+. (V 2 £) 
\/(C 2 — 4BD — C)=£irst approximate root. 

(?=3844. _(_4£x£=+4£Z)=+1512. 

(1512/15356) =73.1846978+. We take the positive root 
because, using the negative root the next use of the for- 
mula becomes imaginary. 
— (— C =(—C ) +73.1846978+ 

(_2£ ( 2B) —62 



42) 111.846978+ (.2663023+ 
(.266323) 3 =.0188842+ 
6^ 

.1133052+ 

(D)*= 18. 

— (18.1133052+) — (Z)) 11 in Formula. 

_84 — -4B 



1521.5176368+ ^4BD 

3844 

V5365.5176368+ —73.249697+ 
Posi tive ro ot = +73.249697+ 
_(_C ) =—62. 



(-2B) 42) 11.249697+ 

Second approximate root= .267849+ 

—83— 



s 4 +V— V both + 

(.267849+) 4 = .005146997+ error x2 

18.1133052+ =/>» in Formula. 

— 18.118452197+^D 111 in Formula. 
—84 =^AB 



+1521.949984548+=45L> 111 
C 2 =3844. 

V5365.949984548+— 73.2526462+ 
Neg. root —73.2526462+ 

— C ==+62. 

2£= —42) —11.2526462+ 
Final approximate root= .267920 + 

In this the approximating root is positive. The sign 
changes from ( — ) to ( + ). The multiplier (2) is not 
used with (A) = (6). 

(1/25) is n\;nus. + root becomes minus affected by 
(1/2B). 

We have above # 4 + 4 y — 1 4 =+1. .005146997+ must be 
multiplied by (2) because it has been subtracted twice. 
We have &*+yl 4 — 2 n \/l 4 =0. 2 n V — l n restores the term 
in the function. This gives a nearer approximation. 

This example must be corrected as the last. 
z 4 +14z 3 +59z 2 +82a*+18. —2±\/3, — 5±\/7. 
A=+14, £=-+59, C=+82, J9- + 18. 
(1 25) [(VC 2 — 4BD) — C]=»first approximate root. 
C 2 = 6724. —4BD— 4248. 
—4248 —82 =— C 
V2476 = 49.7594212185 + 
2B=118 ) — 32.2405787815 — 
First ap. root= —.2732252439 + 

(— 2732252439) 'xH+JD 1 ^/)" 
D=18.18— ( 28555548168) + 

= 17.71444451831+=«Z) 11 

— 4BD —236 

— (4180.60890632116+) 
C- = 6724 

V 2543.39109367884+= 
50.4320443138+ 

— C=— 82.— 82. 

2B=118 ) —31.5679556862+ 
2nd ap. root=»— .2675250481 + 

—84— 



(_.2675250481 + ) 4 =^ 
+ .005122214983+£> 11 =D 1 





+ 17.71444451831+ =Z) 11 




17.71956673329 —Z) 111 




—236 


— 4£ 


= 


—4BD 


=4181.817749057148+ 


4BD— 


4181.817749057148+ 



C 2 = 6724 



V2542.182250942852= 
50.419066343+ 

— C= -82 

25=118 ) —31.580933657+ 

— .267706+=— 2+ V3 approximately. 
_2+V3=^- .2679492+. 
.r 3 — 9tf 2 +17;r+15=0. Roots +5> +2±\/7. 
A=— 9. £=+17. C=H+15. 
Least root 2 — \/7=— .645752+ 
Approximate — .64533 + 



*«* 



Js-etfifinrfy-A 



289=£ 2 
540=— 4AC 



+ V 829=— 28.79236+ 

_ B=+ 17 

— 2A 18) — 11.79236+ +15. =C. 

(_.65513+) 3 = —.2811784 + 
+ 14,7188216 

36 

88 3129296 
441 564648 



529.8775776 
289 =*B 2 



—28.61603+ 
17 



A/818.8775776= —28.61603+ 



18 ) —11.61603 ( —64533. 

x z -\-x- — 41a* — 105. Roots, — 3, — 5, +7. 

x 2 — 41a;=105. Least root numerically -is — 3. 
.r=±l/ 2 (V2101+41) =—2.41833+, first approx. 

—85— 



We take — VW2101. The expression with the positive 
sign would become imaginary. 
—2.41833+) 3 =— 14.14212+ 
2 

—28.28424 
—105 



—133.28424+ 
4 

533.13696 
1681 

-V2214.13696+— —47.05 

+41 



2 ) — 6.05 

—3.02+ 
z 3 +z 2 — 39z+21=0. Roots —7, +3± y 6=5.4492+, or 
.5508+. 

x=± 1 \/(B 2 — 4AC)— £_. A=+ 1. 
2A 2A B=— 39. 

£ 2 =1521 C= 21. 

_4AC=— 84 

V1437=±37.90778+. The negatives gives the least 
B= 39. positive root. 

2A=2.2 ) 1.0922 2 

.54611 First approximation. 
(.54611) '=.1628+ 
2 

C=21. 21.3256+ 3 V— 1 3 =.1628+ 

4 =-4 A. 
_4AC=— 85.3024+ 
B 2 = 1521 

\/1435.6976+=±37.8906+ 
B = 39. 
2A =2.2) 1.1094 

.554 

(.1628), the differential, must not be multiplied by (2), 
ffii *t is (+ 3 V — 1 3 )> and is negative. This gives as result 
.5506 + a nearer approximation to (3 — V6) as above. In 
th 1 '.^ problem the root is taken the least and is positive 
while the differential is negative in effect. 

To obtain the formula on page (163), the (Fx) has been 
developed (n — 2) times success^velv. Each equation above 
the formula is the successive result of series (1), (p, 8). 

—86— 



The first degree equation represents the straight line. 
In the second degree function the determinant of the limits 
is a constant. 

Ins the higher degrees there are two or more deter- 
minants for each limit. So we can use series (1), (p. 8), 
only to (n>2) . For a negative lim£t or root we have 
( — x-\- n \/l). For a positive limit we have (+#+ n V — !)• 
The (nth) root of ( — 1) is always minus. ( n Vztl) is the 
differential unit. For ( — x) and, (n) odd, we have, (+A), 
( — A:r n +A n yi) n , For ( — A), we have (+Aic n +A n Vl) n - 
For (n) even, and ( — x) and + (A) we have +Aa; n +A n Vl. 
For ( — A) we have ( — A£ n +A n yi) n . For (+x) and 
(+A), (n) odd, Az n +A n V(— l) n . For (—A), — Az n + 
A n V— l n ). For (+a;),(+A), (w) even, +Ax»+A"\? (— l) n . 

For (— A), — (— Ao; n +A n Vl n ). 

This determines the use of the multiplier (±2). 

x ^Sx 4 -j-2Sx s — 5l£ 2 +94z+120. Roots —1, +2, — 3, 
+4, —5. 

a;=±l_ 

2CV(£> 2 — ACE)— D . 

2C. I> 2 =* 8836 
4CE=24480 

V33316===t:182.526+ 
_94 

+ 102 ) —88.526 

(—.8679+) 3 X— 23+120. —.8679 



=135.0374 
—204 
27547.6296+ 

8836 



\/36383.6296=±190.744 

94 



102 ) 96.744 =—.9484. 
(_.9484-f) 4 x2.3= +4.854 

135.0374 



139.8914 
—204 

28537.8456 
8836 



V37373.8456= ± 193.3231 + 
94 

102 ) 99.3231 

—.9737 

—87— 



(_.9737+) 6 =+.8752 
139.8914 



140.7666+ X— 204 and add 8836. 
=37552.3864 used logarithms. 

Approx. root =< — .978+. Root is — 1. 

A MORE CONCISE PROOF OF FERMAT'S THEOREM. 

Let (x) be a variable measure, representing all the Tim- 
its of any magnitude, as, a line, or a surface or a solid. 
Let (y) be another measure of all limits of the same 
magnitude, with (x) . (y) will represent all numbers. 
(x) will also represent all numbers. If these measures be 
taken independently, then (x — y) will represent all num- 
bers, and ( y — x ) will represent all the lim- 
its of (y) . (y) is a distance between any 
two limits of (x) , and (x) is a distance between any 
two limits of (y) . Now we will hypothesize, for the above 
magnitude, a line of the (nth) degree, when n>2. The 
ordinate, for points in the line measurered by (x) , will be 
(x — y) n . WJien we assume (y) independent, then (y — a;) n 
will be the different ordinate for any limit of the same 
magnitude. If (y) is taken a magnitude not equal to (x), 
fnd (y — x) n =0, and (x — y) n =0, then the ax/is of (x) 
will be differently placed from the axis of (y). 

The origin is changed by the hypothesis that (y) is not 
equal to (x) . But also when the origin is changed, (y — x) n 
=,( x — y) n numerically. (x n ) = (y n ), for both are numeri- 
cally equal to absolute zero. All numbers are represented 
by (x) and {x) n , so also by (y) and (y) n . In the func- 
tions of the third and higher degree lines, there are always, 
as has been shown in these studies, one or more values of 
(y), for each value of (x) , that can never be = to (x), 
and thjis is so for every limit of (x) in the magnitude. If 
w>2, then these two expressions (x) n and (y) n , each, 
measure every limit of the magnitude and are numeri- 
cally equal while their measuring units (x) and (y), each 
express every number yet are never equal. Then the 
measuring units of these two numerical equals (x — y) n 
and (y — x) n , must be incommensurable. 

Wihen we put the infinitesimal hypothesis upon these 
Vmits, the units become equal, and the two independent 
limits are still incipient-ly independent, when (jr)=absolute 

—88— 



> zero and (y) then = (x) ; or, when (?/)=absolute zero and 
#=(2/)- T ne incipient differential (y) restores the limit 
(x) ; and Fx=the ordinate (x n ) . At the same time, the 
incipient differential (x) restores the limit (y) and 
(Fy), = ordinate (y n ) . In the first case the origin of 
(y) which may be (O 1 ) is the differential origin; and, 
(0), the origin of (x), is the contsant origin, and the 
reverse for differential (x) . Therefore, in any magnitude, 
when n>2, the ordinates (x 1 ) and (y n ) are incommen- 
surable. x n -{-y n is not equal to (z n ) . When n>2, the ori- 
gin (0) is not commensurable with (O 1 ), as to the limits 
of the magnitude measured from the changed origin. On 
pages (49-52) we eliminated the incommensurable unit 
from (Fx) , third and fourth degrees. We thus illustrated 
the process of eliminating the incommensurable unit from 
(Fx) , (nth) degree. As a farther progress in this study 
on the above pages, (49-52), we add, that the fourth 
degree formula at the head of page (50), gives, when all 
coefficients of (x) are constant, a rational, numeral, 
value for (a) and (b) . There are two numbers for (a) 
and two for (b) , giving the four limits when (0), the 
origin, is constant, and (D), is constant. But this formula 
is without (D), and these numbers for (a) and (b) stand 
for all the roots of the magnitude as (D) changes value. 
These numbers are therefore constants, as numbers, for all 
the nositions of the axis necessary to include the entire 
magnitude. The changed value must be in the unit of 
these numbers, which must be compared with the unit 
of the changed origin. If (L) be taken as the variable 
term of one origin, and (L 1 ) for this term of the function 
at the changed origin, then (L 1 ) and (L), being the only 
variables, will have the law of the changing unit of these 
numbers (a) and (b) . 

Now, on p. (72), we have this law of change of the 
value of (x) included in the change of value of (L) to 
(L 1 ) . (y) is the distance from (L) to (L 1 ) . y=i,± (L 1 ) 
— (L). ^y=n^/(L 1 — L)=^y=dif. (x) . 

\/y is the incipient changed (x) in (L 1 ). 

At the top of (p. 72) is the contsant (Fx) . Below this 
(Fx) is the changed (Fx) , now on the changed abscissa 
(X X X X ), marking new limits of the magnitude. These 
limits are measured by a unit incommensurable with the 
corresponding limits on constant (XX). (a) and (b) in 

—89— 



the formula, p. (49), have the unit of the (Fx) on (XX). 
This (Fx) on (XX) is the absolute hypothesis of this 
specific line or magnitude. By making (XX) constant, 
(L) becomes constant, and so a unit is hypothesized, con- 
stant. This unit is the basis for the measurement of the 
entire line, or magnitude. 

Now, returning again to p. (72), we have three differ- 
ent hypotheses of zero. (Fx) on (XX)=zero 1 absolute, 
because of the negative relations of its terms. This is 
always implied when equality is hypothesized. Then this 
same limit is reached when (x), in its unit, is infinitesi- 
mal. It is also reached when (x) is absolute zero. When 
(x) is absolute zero, then its differential (1), the unit of 
measure, becomes zero and gives place to the new value 
"" 7=to changed (x). Lx =L 1 (1) = (£ 1 ) for the new 
value of x. We see that (L) does not give absolute value 
to (x°) or (1) unless (Fx) on (XX) is constant. (x) = 
absolute zero, and so has no differential ratio of incipiency. 
or beginning. (.r)=infinitesimal zero, has a differential 
(1) a varying unit. When (.r)=zero absolute, then ( n \ 
the differential unit takes the place of (x) in (Fx) on 
(XX), and is the new (x) in (Fx) on (X X X X ) with a 
changed value not commensurable with (x) in (Fx) on 
'XX) when n>2. But (Fx) on (X X X X ) will measure 
the entire magnitude and so will measure (x) on (XX). 
This seems a contradiction. Yet we have shown before 
that (X-X : > has all its point sof division on (XX). A 
division on (X X X X ) may approach infinitesimally near co 
a perpendicular on (XX), and yet not coincide as to in- 
cipient ratio. To measure (Fx) we must have first (Fx) 
on (XX) constant. This is done by the formula p. (49). 
We then change the differential unit for all changes of 
(L) and so get roots of all equations of this specific line. 

We wish now to show from p. (72), how to change the 
limits on (XX) to limits on (X'X 1 ). It is there proven 
that the difference between (1) and (2) is the differential 
of (1). (2) includes (1). (2) has the hypothesis of ab- 
solute zero, for (.r)=0, and (1) is absolute zero at each 
of (n) limits. (2) and (1) are then absolute zero. ("\ y) 
is the difference between (x) and the new limit of (x). 
In (2), x=0, a r d ( H > y) takes the value of (x) in (2). 
when (x) in (2) is absolute zero. But (x) in (2) must 
include (x) in (1). Its value in (2) will be (x) in (1)^ 
n W- On p. (72), (*\ >/) is the distance between (x) on 

—90— 



% (XX) and (x) on (X^ 1 ). (y) is the distance from (Fx) 
on (XX) to (Fx) on (X^ 1 ). It will be seen, by the 
above, that limits of (Fx) , when n>2, cannot be olcated 
until the value of (L) on (XX) is found as on p. (49) by 
formula eliminating the incommensurable and getting the 
unit of measure. Take the third degree equation, solved 
on pp. ( ) : 

x z -\-x°- — Alx — 105=0. a:=(a±V&)- By series (1), 
p. (8), 3a 2 +2Aa+£+6=0. 

a=l/S\/ (A°— SB— 3b) — A/3. a=]/3V (1+123— 3b). 

&=1 will make a rational root. 

a=±ll/3 — 1/3= — 4 in the root. 4 will be ( + ) in the 
coefficient. (4-fl)=5 in the coefficient, ( — 5), in the 
root. x= — 3. (+7) is the third limit. 

This is the (Fx) on (XX). Let us find the roots for 
(Fx) on (X l X 1 )=x 3 +x°— Mx— 6. (L— &) = (— 99). n V?/ 
=— 3 \/99- x 1 ^— 3+ 3 y99. x 1 ^— 5+ 
3 V99. ^=7+ 3 V99. 

If we change the signs of these limits and take their pro- 
duct we have the final term — 105+41 3 y99+ 3 y (99) 2 — 99. 
41 3 y99+ 3 V (99) 2 = (—105—99—6) . The logarithms used 
show, by their approximate exactness, that the sum of the 
rationals have the same limit as the sum of the radicals. 

Now ( — 105), ( — 6) and ( — 99) are the variable units 
of (L), (L 1 ), and the differential of both of these (L — L 1 ) 
= ( — 99). These have a different unit or differential. 
When we develop the function by the successively derived 
functions of (L) (L l ). as on pp. (17-20), we develop three 
different functions (L) or (F#)'s. These three develop- 
ments appear separate, when we put the roots of (L 1 ), 
above, in their places as on pp. (17-20). This develop- 
ment always gives the function, as has been profusely 
proven and illustrated in these studies. Three (Fx)'s will 
appear in the constant coefficients, as well as in the final 
or variable term (L), as shown above. In the above equa- 
tion, when the unit of measure is not changed in size, as 
is required when n>2, we will have as roots of the equa- 
tion, 6 and 1/2 ( — 7±3V5). We proved on p. ( ) that 
the unit of (6) here is not the same in size as the unit 
5, 3 or 7, in the original equation with (L)=a( — 105). 
These roots have the same size of number, but not of mag- 
nitude, as when (L)='( — 105). This will be seen by the 

—91— 



reappearance of V) in them. This is not an acci- 

dent, but they always appear in some form in its roots, 
when a limit is assumed on the abscissa and not on the 
ordinate. On p. ({ ), in proof of Format's Theorem, it 
is proven chat while in :: ] and {y) , each may represent 
all the numbers for which the rther stands, and iir) may 
be equal to (:•:-) as numbers, They are never the same size 
as magnitude. 

If we assume a commensurable limit, we are only tracing 

a broken line along the curve touching it at points. If we 
make the new function = ), sind get the limit from the 
new origin, the former unit has ":een changed without any 
law of compa::; m and the roots, (6), and V£( — 7±3\ 5), 
are not rcots of the changed Equation as related to the 
measure used at the origin ::: XX . 

Take the (F;. - ; =.-.-_ 2 .v — 71 z 2 -j-140a: — 46=0. 

Put (x) = a — \ b. Formula, p. 49), we have: 
(b)= — ±£T— 3Aa 2 — 2Bci— C— i 4-— .4.) 
_ __4o _R c 2 — 142a— 14C 4;— 2 

— 4a : — 2 a - — a - — 140a — 140 

4^—145 4a — 2 

4a-— 2 a 

_140«__140 
Put a =(— 3) or (-4). (4o— 2)=— 14 for (a)=— 3, 

or (-14. for (a)=4. &=(— 32) and (—62-. 

U) = (— 3 = \ — 82) and u=\— 62). L=3198. 

The basal eouation for the (Fx) = {: — 2. -71. v — 140.r 
—46) is the equation l — 2 -{-71 .- — U-O.v— 3193). The 
roots of thfs equation are ( — 3 = - — 32) and (4 = \ — 62). 
(L—L = S2U) = (y) when (x) = ( i \ / y). 

The roots sou| re 3 — 3 = \ — 32— -\ 3244) and 

- = - — ■:■! — '-- 3244 . I: the product of these limits = 
absolute zero, then (I/-)=zero, and one value of {x)= 
absolute zero. When (d)=0 and H \ y=(d), then the part 
containing the basal equation above disappears, and we 
have a function in which (L) contains ( — 46) and (3244). 
This sum will = the infinitesimal terms left, when those 
affected by ! =0 drop out. In the third degree example, 
en p. 1 :: was prove 11 , by the values of the i n finitesi- 
mals. that this term has its limit in zero. Then (d) , as 
above, is always zero, and the basal equation falls out. and 
the value of (x) in the example equation remains. 

—92— 



The values of (a) and (b) , in formula, p. (49), may be 
found for (Fx) t (nth) degree. For the higher degrees, 
we have n_k \/&, so that rational (b) will be near a 
lower derived function, and the value more easily found. 
The process will be the same as on (pp. 48-49). This 
formula indicates at once, as in the above example, the 
imaginary roots, when we get ( — 5), when (n) is even 
in ( n yfr)- 




x 3 — 33x 2 +342#— 1100=*). 

Roots 11, 11±V21. 

3a 2 — 66a+342=0. First der. F. 

a=+ll+V7=(00 11 ). 



-93— 



^=-j_ll^_y7=(00 1 ). 

x z — 33£ 2 +342a;— 14y7=Simit P". 
x 3 — 33tf 2 +342x+14y7==rimit pi. 

6x — 66=0. Second dev. F. 

#=11. The value of the second derived function changes 
from (-}-) to ( — ) and the curve line changes from convex 
to concave at (6). In the function of the third degree 
there is one such change . There are two tangents parallel 
to the axis, as there are two limits to the first derived 
function P 1 , P 11 , when the second derived function is ( — ) 
the differential is diminishing in its rate of increase or 
decrease. In the figure above the straight line (ac) is a 
constant differential by which (a) approaches (c). Along 
the concave line (adc) , (a) approaches (c) from the direc- 
tion (d) . In the convex line (age), (a) approaches (c) 
from the opposite direction (g) . (OY) is negative. The 
direction of (a) to (c) in the convex line is negative. In 
the concave line it is positive. This changes when the or- 
dinate is ( + ). The line (aP 1 ) is convex toward (XX). 
(aP) is concave toward (XX). When (x) is less than 
(11), the second derived function is ( — ). At (11) it 
changes and there is an inflection of the line (P^F 11 ) at 
(b). (ef) is tangent to both (P 1 ^) and (P ll b) at (6). 
(Pa) lies opposite to (P xl b). The latter line is concave to 
(XX). So also is (P ll c). In the function of the fourth 
degree the first derived function will have three limits and 
three tangents parallel to XX, P 1 , P'i, P 111 , and two in- 
flections. 

These inflections do not necessarily take place on (XX). 
Wie have the same law for the number of parallel tangents 
and line inflections for the function of the (nth) degree. 
The number of parallel tangents are (n — 1), and the in- 
flections (n — 2). When the limits are all ( + ) or all (- — ), 
the line must be concave to the axis, on that side at which 
it has its origin, until there is a point of inflecion. But 
the line of the second degree has no point of inflection for 
it ssecond derived function is a constant. Also when the - 
sign of the second derived function is the same as that 
of the ordinate the limits of the function are imaginary. 
So that the second degree line is convex to XX. Since our 
solution traces its way to one limit the limit must be found 
on a co n vex line. In Fig. p. 93, we may extend the line 
(P 11 cP 111 ) and put the origin at (Om), and we have the 
same function with signs of limits negative. We cannot 

—94— 



use "the formula (p. 77) to solve the problem on (p. 93) 
from the origin (0), because the curve is concave to 
(XX). We must adapt the conditions to the second de- 
gree function. We must have the origin at (O 1 ) or (O 11 ) 
and so fix the adjacent limits (a) and (b) or (b) and (c). 
We will now change the origin to (O 11 ) and fix the limit 
(b) as a negative limit. If we change the sign of either 
the second term of the function or the fourth term, we 
shall have changed the sign relation of the ordinate to the 
second derived function, and this relation can only exist 
to the right of the inflection (b) . (L), or the variable 
fourth term, is the distance from a limit of the magnitude, 
as P, P 1 , P 11 , to the axis of any limit on the magnitude, 
as (a, b, c) . If we change the sign of (L) we pass to the 
opposite side of the axis of abscissas. If, however, the 
second derived function changes as at (b) , in the figure 
above, then + (L) desnotes a limit of the magnitude below 
the line as (P 11 )- Our second degree equation has its 
origin at (O 11 ) and traces the line (P 11 ^) to the limit 
(b) , and (L) is the variable limit of a second degree mag- 
nitude changing its ordinate till it fixes the limit (b). 
We have (OO 11 ). If we subtract (0 ll b) from this we 
have (b) from (0). The solution is as follows: 

(z 3 — 33a; 2 +342:r+1100) . 

A =—33. £=342. C=1100. x= + 7 66 7 [ (342) 2 — (— 4x 
33X1100)]— 34 7 66 =— 2.576+. (— 2.576+ ) 3 =(— 16.99+) . 
—16.99x2=33.98— 1100— 33.98=— 113398+. —1133.98+. 
Change the sign as before, and, ( — C), or (L)= — (.113398 
+ X— 132) =+149685.36+. Add 116964=£ 2 and extract 
root. Change the sign of — 342 because the division is 
minus, and use as before the negative root and we have 
(—2.64) the limit (b) from (O 11 ). Il+y7— 2.64+=+ll, 
the limit (b) from (O). 77 — 2.6457. The decimal must 
be carried out farther for greater accuracy. 

The equation at zero for the limit (P 1 ) from (0) is 

x 3 — 33£ 2 +342z+14y7. When we have x 3 — 33o: 2 +342^+ 
1477=0, then the tangent at (P 1 ) is the axis and (O 1111 ) 
becomes the origin. The limit (P 11 ) is (x 3 — 33x 2 +342.t — 
1477). (11 — \/l)=x substituted, gives the first, and 
(11+77) =# gives' the second. 

These ordinates differ only in sign. Let (L) be any 
point on these ordinates. Then the equations will be alike 
except in the sign of the ordinate. If th sign -of the root 

—95— 



in either be changed it will give the ordinate of the other 
ordinate. The roots therefore differ only in the sign. And 
the two curves are alike but reversed. 

If we change the origin to (O 111 ), the line will be the 
same from (O 111 ) as from (0) with signs of limits 
changed. We then have ar+33a; : -'+342a;+1100=0. If we 
change ( + 1100) to ( — 1100), and proceed as before we 
get the limit (b) from (O 1 ) beyond the inflection from the 
origin now as before. We get (-(-2.64+). Take the general 
equation of the second degree and find the first and second 
derived functions and we have, — 

x s -\-Ax 2 +Bx+C. 3a 2 +2Aa+#=0. 
x 2 +2/3Ax=*—(B/S). 

£=l/3A±l/3 V (A 2 — SB) . 

Also, 6a;+2A=0. x=* — 1/3A is the point of inflection. 
On either side of the point of inflection will be a like line 
as above shown in the specific equation. The lines will 
not.be symmetric with the origin, but reversed. The in- 
flection is not necessarily on the fixed axis. It is at a 
tixed point on the line by its law of coefficients. 

This figure does not represent proportionate lines. Only 
relations to the axes are illustrated. 

£ 3 +24a; 2 +129:r+170=0. 

3£ 2 +48£+129=0. 

z 2 +16a?+43=0. 

s=i_8±Y(— 43+64) 

x=— 8h=V21. (P 11 ? 1 ). 

6z+48=^0. x=— 8. 

Point of inflection is ( — 8). 

What has been shown in example on (p. 93) is true of 
this, except, that this illustrates the instance where the point 
of inflection is not on the axis of absdissas. 

If we substitute for the points (P 1 ),^ 11 ), the values of 
the ordinates of these points, we have — 8+42\/21 as dis- 
tance from (P) to (P 11 ) and — 8 — 42 a 21 as distance from 
(P) to (P 1 ). (P) is the fixed point from which the dis- 
tances on the axis of ordinates is computed ( — 8) locates 
the axis of abscissas (dh). From (e) the line repeals 
itself to the left toward (P 111 ), but the points are in re- 
verse order. The differential of (6.T+48) is (6) and is 
constant . When (.r) reaches ( — 11), then the point de- 

—96— 




scribing the line has the same movement negatively with 
which it appoarched the point (e). It moves therefore 
constant along (ed) as it was moving before toward (eh). 
The differential of the function is the same increment. 
These causes retrace the line (eP) in the direction 01 
(P 111 ). The points are symmetric but reversed. 

If we substittue (x — 17) in the equation (# 3 -j-24# 2 -|- 
129^+170), we get (x 5 — 27x 2 +180z). This removes the 
origin to (C) and the roots are now (-f) and mark the 
same limits on the line as before. The first derived func- 
tion is {3x 2 — 54^+180). The parallel tangents are at the 
same points as before. The limits of these parallels are 
(9d=V21) measured from (C). Substitute (9q=y21) j n 
the new equation and the ordinates of the tangents are, as 

—97— 



before, (162^-42y21). The point of inflection is (9.) as 
before. This curve is symmetric as to the two loups be- 
tween (P : ) and (P 111 ). If we substitute {x — 19) we get 
z 3 — 33ir-+30(hr — 4 76 The point of inflection is then +11. 
The limit (C) i3 now (2) from (O 111 ). The limit (b) = 
4-14, and (a) =+19. The line (PP 1 ? 1 P 111 ) is measured 
reversed and lies symmetric between (0 and O 111 ), so that 
when the law of change in the first derived function 
reaches its limit at the point of inflection (e) and traces 
a line identical with that from (e) back to the origin. If 
we substitute (x — 8 — \/21) in the function (£"+24a' 2 -h 
129*+170=0), the result will be .r 3 — 3y21a; 2 ± (0)+162+ 
42\/21=0. The point of inflection will be +V21 as be- 
fore. The tangent point (+2\ / 21), which marks the 
point (P 1 ) from (O 11 ). The limit marking a point of in- 
flection cannot be commensurable with any other limit of 
curve except points on axis of abscissas. When origin is at 
a point of inflection we shall have in the third degree 
(6a=0) (A=0). When the origin is at (O 11 cr O 1 ), 
77), the roots are ap. ( + 1.40+) , correct (1.475), — ap. 
(_1.567+), correct (—1.5825+). 

The equation is (x z ^-S\ / 21x 2 ± : (0) +162 — 42\/21). 

When the origin remains at (0), the successive derived 
functions remain the same for all points of the curve. The 
differential law is not changed. The differential law 
changes with the origin, yet it traces out the same line 
or succession of limits as before with the same law gov- 
erning the varying (L) or last term of the function which 
does not change save with the change of the axis of ab- 
scissas. A, B, C, etc. change numerical value with the 
change of the position of the axis of ordinates. (L) 
changes with the change of the axis of abscissas. The 
values of (x) are different in the changed origin, but the 
differential result remains the same (L-\-y) as it traces 
the same limit as before. (3*T 2 +48a:+129), with its value 
of (x) , will give the same result as (3x 2 zt6\''21x) when 
(x) denotes the distance of the common limit from the 
changed origin. 

On (p. 77), it is proven that all the functions, (1, 2, 
3, 4, etc.) have the same differential (at zero), or the same 
value for (y) , (L) being constant, in the different func- 
tions. Functions (1, 2, 3, etc.) differ from each other m 
the differential unit "( n \ y, n \ 7 y n r\ etc.). In the formula, 

-—98— 



(p. 77) /the origin is not changed. In this formula each 
differential function (1, 2, 3, etc.) includes all that pre- 
cede. (2) includes (1) at zero. (3) includes (2), and so 
must include (1). The last one must include all the pre- 
ceding ones. To make this clear the terms omitted in the 
formula must be substituted so that (n — 1) will be 

n V(— l) n +A n V(— l) n - x +£ n V— l) n - 2 +C 

n V(— D n - 3 

+ e /x 2 +A~£+L=0. If we take the line on (p. 58), the 
above function includes all its limits. x=( n \/ — 1) is a 
measure of all the limits. 

The value of ( n V — 1) as a number covers the entire 
function. (Jx 2J r Kx-\-L) is a line of the second degree. 
With the changing values of (x) its limits include the 
function of the (nth) degree. It has been proven that the 
function of the (nth) degree has (n — 2) points of in- 
flection. It has been shown that in the function of the 
(3d) degree the line has one point of inflection and at 
this point the curve begins to repeat its proportions in a 
reverse order. There is one portion from each of two 
second degree curves. The common abscissa passes 
through the point of inflection. See (pp. 93-95). A 
(4th) degree line would have two points of inflection and 
has two loups on one side the axis of abscissas and one 
opposite. After a point of inflection thse differential will 
retrace itself. 

Take the Ex. (p. 93), (x 3 — 33x 2 +342x— 1100). 6a;— 
66=0 is the point of inflection. When the point (+11) 
is reached then the rate of differential change is zero. 
The rate began with (x) at zero and approached to (+11 
— 11). The rate now begins at zero and passes on to (11) 
through all its changes but in the same direction, which 
reverses the magnitude as to its origin. 

When (9—11), becomes (13—11). The first is (—2). 
The second is (+2), and is the same rate but opposite in 
effect. The rate of differential change is constant at the 
point of inflections and marks the direction of the tan- 
gent line. The incommensurable unit passes to the com- 
mensurable here. Take the fourth degree Eq. x*-{-6x* — 
2l£ 2 — 62o;+18=0 (p. 83). Second derived function is 
(12x 2 +36x— 42=0). x=V 2 (— 3±V23). 

z*+6z 3 — 21z 2 — 62z+18. Roots ( + ), ( + ), (— ), (— ). 
Points of Inflection (— 3±V23). 

—99— 







The above figure may be referred to as describing the 
general elements of the line referred to in the pages that 
follow. (MM) and (P) represent portions of a line of the 
(nth) degree, (tt) are tangents passing through (0) the 
natural points of origin. These points are described by 
(F 2 x)=0. (P) represents extreme limits defined by (F x x) 
=0~. (XX) (Yi), with (0) as origin, represent rectangu- 
lar axes measuring the line from a point of inflection, 
which is a natural origin of the line. (X X X X ) (Y^ 1 ) rep- 
resent rectangular axes, with (O 1 ) a changed and general 
origin. (X'.'X 1 ') (Y 1T Y n and (0 ) represent axes with 
acute or obtuse angle. These elements of the line are not 
placed in relation to each other, as they are intended only 
to represent the general line. 



We desire now to show the dual nature of the line or 
magnitude whose content and limits are expressed by the 
function of the (nth) degree. Series (2), (p. 8), shows 
series (1), (p. 8) to have, as its content, two functions 

—100— 



numerically identical, (Fx) and (Fy) . These functions 
have opposite signs, and therefore denote infinitesimal 
and numerical departure in opposite directions from the 
origin. Also (x) and (?/), the differential units, have op- 
posite signs, and so denote departure in opposite directions 
on the axis of abscissas from the origin. Therefore, for 
every limit of the line or magnitude in (Fx) , there is a 
corresponding limit in (Fy) lying negatively and identi- 
cally situated and described from the origin. Therefore 
every (Fx) has for its content two identical magnitudes 
negatively situated from the natural origin of the line or 
magnitude. This is true of (Fx) because series (1), 
(p. 8), contains the limits of (Fx) when (y) and (Fy) 
is at zero; and the same, when (x) and (Fx) is at zero. 
(x) is the differential of (y) and is always equal to (y) , 
and (y) is the differential of (x) and is = (x) . (x — y) 
is a limit of (x), and (y — x) is a limit of (y) . These 
are equal and have contrary signs and are the same as 
limit (x) or limit (?/), or differe n tial (x) , or differential 
(y) . All lfmits in the figure of a line or magnitude are 
contained in the one differential departure from the ori- 
gin, which, at once, by an absolute hypothesis of size of 
unit or differential, reaches each or any fixed limit, with- 
out progress through space or time. (Fx)=0 hypoth- 
esizes the limits of a line or magnitude. Put (L) con- 
stant, and we make the absolute hypothesis that gives size 
to each of (n) limits, or, units of measure. The unit jf 
magnitude takes the form of number which relates X 
comparatively to the other (n — 1) units of magnitude of 
(Fx) when (L) is constant and denotes a fixed origin, 
(0). 

As has been shown before, n \/y, or n \/l, is the unit of 
measure, the differential, and =(x), and (y) is the dif- 
ferential of (L). 

We wish now to prove that (Fx) is a dual magnitude 
from the usual form of (Fx)=)X n -\-Ax n - 1 -\-Bx n - 2 .... 
Jx 2 -\-Kx-\-L — y. If we change the sign of the differential 
unit all the even powers of these terms will retain their 
signs and the odd powers will change signs. Now, if we 
subtract the funtion formed by negative differential 
( n Vl) from the above (Fx) the even terms will disappear 
and we shall have — 2 times the sum of the odd terms 
+2y. This new (Fx) will always be the differential dis- 
tance on the axis of ordinates from the limit indicated by 

—101— 



the first (Fx) to the second (Fx) . The factor ( — 2) is 
the difference between differential ( — 1) and differential 
(+1), using ( + 1 as subtrahend. The distance from 
the origin (0) will therefore be ( — 1) into the odd power 
terms ( — y). This limit is now, as to the axes and origin, 
negative to the limit indicated by the opposite differential. 
If we had subtracted the (Fx) in a reverse way we should 
have had (+y), locating the differential limit on the op- 
posite side of the origin, the limit indicated by (+1) and 
(—?/). The distance indicated is a differential or infini- 
tesimal distance lying between the two (Fx) . These dis- 
tances are general, identical numerically, and opposite in 
relation to the axes. For every limit of (Fx) there is a 
corresponding limit negatively relatsed to the axes. 

There are therefore two lines or magnitudes identical, 
inversely located, in (Fx) . 

We wish now to find the natural origin of the line or 
magnitude, the law that fixes the points of beginning. 
Also to find the law that describes all ratios of a change 
from one limit to another limit. 

The first derived function of (Fx) or, (F x x) denotes ill 
rates of change to include the whole line or magnitude. 
If the (F r x) be made constant then the curve becomes the 
tangent line at this limit. The (Fx) at this limit is the 
law of the straight line. Now when (F ± x) is constant, 
(F 2 a0=absolute zero. Now if we put (F 2 $c)=0, then, for 
those values of (x) that render (F 2 x)=Q, we shall have a 
point of the line where the curve and its tangent are the 
same line as to the law of lines. 

Now, the changes of (F 2 x) have depended, by the above 
hypothesis, on absoltue values of (x) , not on differential 
values. They take place at limits indicated by roots of 
(F 2 x). Let us change the origin to this tangent poi n t, 
then the root or limit of differential (x) becomes absolute 
zero. There is, of course, a new and changed (Fx) , also 
a changed (F 2 x). But these changed expressions co n tain 
the limits of the first (Fx) and (F.x) and describe ah 
the limits cf (Fx) and (F 2 x) from the new origin. We 
must bear in mind that the hypothesis is that the (F/x) 
is constant. This is not so when the origin is taken at 
other points of the curve. The differential rate of (x) 
is constant, (1). (F x x) is also constant at this limit. 
Now if we do not give a negative value to differential 

—102— 



> (1), all its values will express limits of the line already 
described to the tangent point of origin. If we give neg- 
ative value to the differential (1),. the tangent line will 
be on the opposite of the new limit. But the new limit 
is a differential limit, and is any limit, and is numerically, 
by absolute hypothesis of magnitude, any limit, and so on 
either side of the changed origin where the (F x x) is con- 
stant we have identical magnitudes. 

As there are (n — 2) roots to (F,x), there will be 
(n — 2) points in the line (Fx) , where (F x x)\is constant 
and the tangent changes to the opposite side of the line 
(Fx). There will be (n — 2) identical curves in the (Fx) 
of the (nth) degree. 

In the above hypothesis we have Fx=x n -\-Ax n - 1 ~\-B-,\; '- 2 
. . .■ -\-Ix*-\-Kx=§. (L) is zero absolute. : ; 

Also, (F 2 x)=n(n — l)# n - 2 + (n— 1) (n— 2) Ax n ~ s ... . 
6Ix=®. (Fx) includes lines of the second degree. ( — 1) 
and ( + 1) ar e as differential magnitudes, measures of op- 
posite identical parts of (Fx). From any origin the dis- 
tance between them is (±2), the units being necessarily 
equal only at infinitesimal zero. The 11 x ti -\-y n =^(x J ry) n - r 
(y-\-x) n . We "have ± 2x infinitesimal (1). But the factor 
(2) is not an (nth) poweri The Curve of the third and 
higher degrees include this change of the unit from ( — 1) 
to ( + 1). As the third degree has one point of inflection 
where the line changes to the opposite side of its tangent, 
and, so involves the change in the infinitesimal unit. The 
higher degrees increase these changes. (x n ) and (y n ) 
as symbols of magnitude, when (x) is the differential of 
(y) and (y) is the differential of (x) , cannot have a com- 
mon unit except at absolute zero. 

Therefore their sum can have no common unit with this 
above hypothesis, and could not be (z n ) . If, on any hy- 
pothesis, that includes all the limits of (x n -\-y n ), their sum 
is not a perfect Ozth) power, then, their sum is not a per- 
fect (nth) power on any other hypothesis. For if two 
hypotheses include wholly the same thing, then, they agree 
with each other touching that thing, and Fermat's Theo- 
rem, (# n -f y n ) is not equal to (z n ) when w>2, is established 
as a fact. The hypothesis that (x), is the differential of 
(y) a n d (y) the differential of ix) t . above the econd de- 
gree has been establish^ »§m^ beyond 
an exception that, 4temeli*des rffTMtMJ w,hen n>2. The 



Theorem of Fermat is a frequently repeated corollary of 

".'. t ; t =r.;:::es 

~ by its limits, marks all the rates of changes that 
t" sh the ratios between limits as measured by the 
ixe.5 Et describes the different angles that the changing 
::::t:: '.Liz ~i:-^ ---':_ ::r :.: s :: a':s::s5i5 "":.^:. ~.'i:s 

Fa =0, absolute, then the angle is zero and the tangent 
line :s parallel to the axis of abscissas. Then, at this point 
the line again approaches the axis of abscises There are 
(» — 1) roots for F 1 and so {n — 1) points that limit 
the space above and below the axis of abscissas within 
which ~ lies, The final limit of the curve will have 
no point of inflection and so be a second degree line of in- 
definite length. 

The magnitude exists before its measurement. It can- 
not be defined without an origin of measurement a dif- 
ferential unit, (x) of measure and (Fx) to denote its can- 
't ;.: and limits. The origin I may be chosen at liberty. 
tf axes are used they may be at any chosen angle. These 
together will adapt the values Fx i to all points of the 
line or magnitur.T ~ • is then the law of the magnitude. • • 
We have already shown how the origin (0) is changed 
on the same axes rectangular). When (0) is changed 
along the line of ordinates, then (L) is variable. (L — y) 
is variable (L). (y) is the differential of (L) and is equal 
to i ± L When the origin changes along the line of ab- 
scissas then the terms, A. B. C. . . . K are variable 
~.-mi They are not constant numerically as before, yet 
the changed origin and the new {Fx) includes the magni- 
tude as before. (F ■ defines the same results in rates of 
change. iF,x) in the new function marks the same points 
of natural origin. If we were to change the angle of the 
?-xes " -z should chancre (F v x) as denoting a change of an- 
gle for the tangent that touches every point of this same 
line. Then (Fx) would be changed to adapt it to the 
changed ( F. x) . F.x ) would also change. Yet the line or 
magnitude could be measured by the new method of meas- 
urement. If a oolar measurement be used, the tngnomeTi- 
cal line will be the differential measure, and the radius vector 

Pa . If a ime of a degree ji>2 be supoosed to be meas- 
ured or traced by radius vector, then [Fx) will have all 
the properties ihat have been devefoned in these stud: : ; 
The method of measurement cannot change the properties 

:' F =: In all mea3»ement there will be a differen- 
tial unit measure and (Ft)-=D, a limit. 

—104— 



VARIABLE MEASURE AS A UNIT. 

In these studies, (x) and (y) are taken to represent the 
distances of any two independent limits from the point of 
origin, or point from which all limits are reached when 
the law of the magnitude is given. 

They do not at first represent a number of units, but 
each symbol separately stands for any limit. So each in- 
cludes all the limits of the other. In any hypothesis their 
only common limit is infinitesimal zero, or equality, the 
same limit. Absoluteezero is constant, r Infinitesimal zero 
is variable. Infinitesimal zero includes absolute zero. The 
variable hypothesis includes the constant. When the con- 
stant hypothesis is assumed and the variable hypothesis 
ceases, then, the limit (x) or (y) becomes a number. When 
(x) becomes absolute zero, then differential (y) or infin- 
itesimal (y) includes (x), and so all values of (x). When 
(y) becomes absolute, or constant zero, then differential 
(x) includes (y) and all values of (y) . If one of the two 
limits, above represenetd, be taken equal to the distance 
between two limits of the other symbol of limits, and the 
other symbol of limits, at the same time, be taken equal to 
the distance beeween two limits of first mentioned symbol, 
then these distances between limits of one symbol will be 
equal to the distances between limits of the other symbol, 
but with negative relation. So that each symbol is the 
differential of the other, and the differential is negatively 
related. Referring to the figure, (p. 100), let (x) and 
(y) be limits to the left of the origin. Measure off from 
the left of the limit (x) a distance equal to (y) , assuming 
(x)>(y). Then (x — y) is a limit of (x) . Now measure 
off to the left of the limit (y) a distance equal to (x). 
Then (y — x) is a limit of (y). But (y— x) = (x — y) with 
negative relation, (x — y) is a distance to the left of the 
origin (0). (y — x) is a distance to the right of the ori- 
gin (0). These distances from the origin are equal. Any 
magnitude measured as above will have two identical parts 
related to the origin. The hypotheses of (x) and (y) as 
above are universally true of any (Fx) or (Fy) . Therefore, 
every law or (Fx) must have two identical parts as mag- 
nitudes. The above hypothesis is not true except whe n 
(x) and (y) are independent variables, for thisfis a part 
of the hypothesis. One of the two symbols must be con- 
stant, with (Fx) im the second degree. 

If we put the symbols (x) and (y) on opposite sides 

—105— 



of the origin, (O), and make (x) equal to (y), then (x) 

— ( — y) becomes (xj{-y) , an^ not absolute zero. (-—2/) 

— (+x) = ( — y—x). These limits are numerically the 
same and negatively related as when (x) and (y) were on 
the same side of the origin. But now we cannot hypoth- 
esize absolute zero or constant zero. In finitesimal zero 
here does not include constant zero. When we put the dif- 
ferential equal to its variable limit we do not have abso- 
lute zero, but, infinity, (y) is -not within the limits of 
(x), nor (x) within the limits of (y) . : In itheuse -of 
coordinates, in these studies, (x) is the axis o£ abscissas 
and (Fx) the axis of ordinates. (L) is the product of all 
the differential limits, and is the symbol (y). All limits 
must be within the axis of ordinates or the limit -will be 
imaginary. This has been shown in the measurement of 
the hyperbola.; In general an imaginary quantity is an 
improper sign of the differential. 

In the above theory, (x—y), (y — x) and (x) or (y) 
may be absolute zero and infinitesimal zero. ^ 

This is a common limit for all magnitudes. When these 
two limits, absolute zero and infinitesimal zero, are .hy- 
pothesized, then all limits and all magnitudes are included. 

(x — y) is a differential limit of (x) . When (x==y), 
then the differential liniit (x — y) 'is, at the origin and rep- 
resents (x) at zero. If we make (x) constant it will be 
absolute zero, and (y) is by hypothesis equal td (x). (y) 
also as an infinitesimal, or differential, or incipient of (x), 
includes both (x) at absolute zero and at the limit (#). 
For, since (y) is a limit in general, the infinitesimal is 
any incipient constant, and so, may on the infinitesimal 
hypothesis, be any value of (x) hypothesized. So that, 
both on the absolute and infinitesimal hypothesis, separ- 
ately, (y) becomes the limit (a;). In accordance with the 
above theory of the differential limit, (x—y), when (yy 
is the incipient or differential of (x), and (y) represents 
(x) both at zero and any limit, when (x) equals absolute 
zero, — all the conclusions of these studies have been 
reached. ; ; . 

J. '■:.;■ 

'"'- In the equation of' the (wth) degree, x^= n \ / l, n yi is a 
variable limit, which approximates incipiently a limit near- 
est the origin, when (y), or (L), becomes constant. It 
has been shown in these studies that all the coefficients of 
(Fx) are the successive derived functions of (h), the 

—106— 



final term; and, when (L) is hypothesized constant, then 
these coefficients become fixed as to the (n) limits. 

The differential series, (p. 8), can define only one of 
these limits at a time. With the coefficients A, B, C, etc., 
variable, no definite magnitued is represented. 

In these studies, only two independent variables, (x) 
and (y), are used. If we add (z) , then (z) would include 
the differential limit, {x — y), the same as (y) includes 
(x) . But, if (z) is the differential of the limit (x — y), 
it is also of (#). 

Any number of variables may be taken independently to 
compass the same limits. They will each include absolute 
zero. Differential (x) ^differential (?/)=differential (z), 
etc. Each represents the one general limit, (variable 1), 
which is the same magnitude at absolute zero; and each 
becomes the other when the one limit, fixed by the differ- 
ential series (1), (p. 8), is reached. 

The variables (x) , (y), (z) , etc., are not numbers. They 
are each variable (1). Number is the last and constant 
hypothesis applied, as has been shown above. By hypothe- 
sis they are equal, only when they stand for the same limit 
zero. When (x) is zero, (y) stands for any limit of (x), 
and the reverse. (l) = (x), becomes equal to, (l) = (y), 
when (l) = (x) becomes absolute zero. That is (1) = (&), 
cannot =, (1) ='(?/), for any appreciable value of (x) , or 
(1). There is therefore no common measure for these 
magnitudes (1) and (1), but an infinitesimal or approxi- 
mate one. So that with the hypothesis that (x) is the 
differenaial of (y) , and (y) of (x) , (x n ) and (y n ) have 
no common measure. If we put (x nJ r y n ) = (z n ) , then >ve 
must have a common measure in (z n ) for (x n ) and (y n ). 
(x n -\-y n ) is not equal to (z n ) when (x) and (y) are inde- 
pendent variables. In all laws of magnitudes above the 
second degree, they are independent variables. In all 
laws of magnitudes above the second degree, they are in- 
dependent variables. (x n ) commensurable with (y") is 
impossible in (Fx) or (Fy) when n>2. 

If x, y, z, etc., be independent variables, measuring dis- 
tances from the origin; then (x — y) may be taken as the 
differential limit of (x) , and [(x-^-y) — z] may be taken as 
the differential limit of (x — y) . As before discussed in 
these studies, (y) will be the differential of (x) , and we 
shall have the series (1), (p. 8) .:— Fx+(F,x) (F^y) f 

—107-^ 



(Fjt) iF-_-i/) — (F : x) (F^ 3 y) . etc. This is the differential 
function (x — y) a . It is also the general differen- 
tial I Fx I . Now. if we make this differential function, 

— . ), a limit to approach from limit \_(x — y) — z], and 
so develop series (I), (p. 8), we have: F.x—F.x.—a. 
series with (Fy)=ta identical zero. (Fx, y) becomes ab- 
solute zero. We then have (F.x) (F-_ : : ) — (F.x) (F r . z). 
We have the original {Fx) with (2) as differential, (y) 
with (.r). has become absolute zero, {x — gf)=0. sc= ab- 
solute zero. Infinitesimal (y), which took the value of 

'. now becomes absolute zero, {x — y) is constant at 
zero, (z) takes the place of (y), as (y) did the place of 
(x). (z) is the infinitesimal or incipient value of (.*:), 
and when constant is the limit (x). We see, by differen- 
tiating series (1), (p. 8). that that portion of the series 
including (y) and (z) , isolates from (Fx) including only 
(z). This can be illustrated by using, for (Fx) , the nu- 
merical function of the third and fourth degrees. (Fx) 
will reappear, and the sums of the terms containing (y) 
will be zero. This shows that, if (y) . (z), etc., be taken 
as infinitesimals of the first, second, etc.. orders, only 
one infinitesimal exists at the same time. Each order fully 
represents the variable measure and so is always 

equal to every other order. This must be so, as only cne 
limit can be hypothesized as constant at the same time. 

(Fx), in the function of the second degree, equals (; : — 
Ax—B). When (x) is variable, it will have two limits for 
(B) contsant. (y) is the constant distance between these 
limits. If (y) be variable, (x) will be constant, (x — y) 
represents the second, and only remaining value of (x) , 
and (y) has one value. 

Series (1), (p. 8), for the second degree is Fx — (F r) 
I F, .y i - ( F,__;. 1 . Differentiate and F 1 x—F 1 x—(F U _ y) -f 
a constant, or (1), variable. This series is zero, and 
(F- ./) is (y) and (y) is = a constant in the series, x — y 
is absoltue zero when (x=*§) ; and, (x) becomes constant 
at zero, (y) will then become variable and differentially 
and incipiently be the limit (x) . This is the 
same as the reasoning with (x) and (y) inde- 
pendent variables ; except, in this case, in the second de- 
gree function, (y) is not variable nor the differential of 
(x), until (x) becomes constant. In the (Fx), first de- 
gree, (y)=*0. In the second degree (y) has one value, 
and so. constant. In the third, and higher degrees, the 

—108— 



variable power of (y) begins and increases with the de- 
gree of the function. 

The conic sections are measured by (Fx) second degree. 
The parabola is an example of (Fx) first degree. The 
ordinate =\ (2px) . If the axes be changed, and (x) be 
the ordinate, then x=*(y 2 /2p) . If we continue to use (y) 
as the differential, and (x) as variable, as elsewhere in 
these studies, we may represent the ordinate in the chang- 
ed axes, by (z) . Then z=(x 2 /2p). In this case the func- 
tion can never be absolute zero, (x — y) may be hypoth- 
esized, but the function fixes no constant limit, r — r 1 is 
always constant at zero. There is only one variable meas- 
ure (r). This alone must measure the coordinates. 

It has no limit but absolute zero. In the ellipse (r-j-r 1 ) 
=2A, a ccnstant lmit. (r — r 1 ) has the limit zero, abso- 
lute, and infinitesimal. When (r) is zero, absolute, then 
(r 1 ) is variable and = (r), and two limits are located when 
(x 2 -\- Ax-\-B-\-y)=0. This locates the points in the line of 
the ellipse or circle, two limits, similarly situated for each 
hypothesis of value for (y) . When we have (x — \/y) as 
differential limit. In the hyperbola, we have (r — r 1 )^^ 
constant, and never equal to zero, absolute, (x — y) can- 
not be hypothesized to be any limit, general, where such a 
relation exists. When (r) = absolute zero, then (r 1 ) is 
not = to (r), but = a constant. But, in every (Fx), 
(x — y) does exist as a limit of (x) , in which when (x) 
is absolute zero, (y) is = to all values of (x) . Therefore 
(Fx) in the hyper bola must have no real limits. 

If (A) = one-half the axis, and (B)=the ordinate; then 
( — B 2 ), in the hyperbola, is the square of the ordinate 
(L), so the symbol (y) is imaginary, \/( — y) • The line 
does not lie within the limits of the ordinate. If, in the 
third degree function and functions of a higher degree, 
limits do not lie within the ordinate they will be shown 
by imaginary roots. This has been proven elsewhere in 
these studies. The differential, in the hyperbola, 
has the same sign of the variable. There is no limit 
of absolute zero from positive and negative relations of 
terms of the (Fx) , supposed. (Fx) is not real, but imag- 
inary. 



■109— 



IN'DEX 

and 

GUIDE TO THESE STUDIES. 

These three hypotheses appear together as the tissue of 

the body of the entire work. They are the only elements 

in every step of the logic, — used always — 

(1) (x) and (y) , alternately, absolute zero 

(2) {x) and (y), alternately, infinitesimal zero, 
variable (1) as differential 

(3) (Fa;) and (Fy) , [( + ) and (— )] zero, or 
limit zero 83, 90 

(1) The differential ratio, (nx*- 1 ) 6. 7 

(2) General proof, (nx*- 1 ) 25, 63 

(3) Proof by Combinations 28 

(Ax) and (Ay) referred to 8, 9, 10, 15, 16 

Element of time eliminated 30, 31, 32 

Fermat's Theorem 23-25 

(1) By two incommensurable units in Parabola .... 41 

(2) By increment of constant (x) and (y) 44, 45 

(3) By law of combinations 45 

(4) By incommensurable unit 61, 88, 39 

(5) Differential proof 56, 57 

(6) Second degree and third degree Equations ...... 14 

(Fx) =0 contains all limits 3 

Taylor's Theorem 10 

(Fx) and (Fy) , — Analysis of, 11, 17-21. Entire book 

(Lx°) developed into (Fx) , 

17-20, 25-27, 33, 41, 42, 45, 46, 81, 82 

Integrals ; — Circle, Sphere, paraboloid 12-15 

Integration of conic sections 33-41 

Integral formulae 34-36 

Discussion of conic sections by series 1, (p. 8), 67-71 

(1) Solution of Equation of the (nth) degree by 
approximation, and examples solved . . . 72-82, 83-88 

(2) Solution of the equation of the (nth) degree 
by first eliminating the infinitesimal, and find- 
ing the measuring unit of the specific basal 

equation of a (Fa;) . . . . 48-54, 57, 58, 89-93 

Meaning of the unit, or differential (1), 5,16,17,71,80 
General discussion of (Fx) and the line, inte- 
gral law, same as differential, — see Index. 

The (nth) order of the differential, [(x — y) 

—z]—n, etc 107, 108 

Ratio, or number, as the differential unit 64 

LINOTYPED AND PRINTED BY THE COREY PRESS, ENID. OKLA. 



. 17 I"" 



One copy del. to Cat. Div. 



OCT 17 Ml 



